{"id":6891,"date":"2026-09-13T21:07:06","date_gmt":"2026-09-13T15:37:06","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=6891"},"modified":"2026-09-13T21:07:12","modified_gmt":"2026-09-13T15:37:12","slug":"class-9-maths-chapter-6-measuring-space-perimeter-and-area","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/class-9-maths-chapter-6-measuring-space-perimeter-and-area\/","title":{"rendered":"Class 9 | Maths | Chapter 6 \u2013 Measuring Space: Perimeter and Area"},"content":{"rendered":"<div class=\"mymoc-top mymoc-entity-placement\" id=\"mymoc-3644894657\"><div id=\"mymoc-2869687770\"><a href=\"https:\/\/amzn.to\/4ehRB6n\" aria-label=\"printers\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/printers.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/printers.png 1303w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/printers-300x73.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/printers-1024x250.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/printers-768x187.png 768w\" sizes=\"(max-width: 1303px) 100vw, 1303px\" width=\"1303\" height=\"318\"><\/a><\/div><\/div>\n<h2 class=\"wp-block-heading\">End-of-Chapter Exercises &ndash; Complete Solutions<\/h2>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">1. Intro<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">This chapter develops the idea that <strong>area and perimeter can be understood through decomposition, rearrangement, similarity, scaling and circular geometry<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The End-of-Chapter Exercises bring together:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Algebraic identities using area models<\/li>\n\n\n\n<li>Areas of triangles and quadrilaterals<\/li>\n\n\n\n<li>Heron&rsquo;s formula<\/li>\n\n\n\n<li>Circumference and distance travelled by wheels<\/li>\n\n\n\n<li>Areas of circles and quadrants<\/li>\n\n\n\n<li>Trapezium and kite formulas<\/li>\n\n\n\n<li>Scaling of shapes<\/li>\n\n\n\n<li>Shaded-region problems<\/li>\n\n\n\n<li>Circle packing<\/li>\n\n\n\n<li>Geometric dissections<\/li>\n\n\n\n<li>Pythagorean relationships using semicircles<\/li>\n\n\n\n<li>Advanced area relationships<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Unless stated otherwise, the textbook asks students to use<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>&pi;<\/mi><mo>&asymp;<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\pi\\approx\\frac{22}{7}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">2. Learning Outcomes<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">After completing these exercises, students should be able to:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Represent algebraic identities using geometric areas.<\/li>\n\n\n\n<li>Calculate areas of triangles using suitable methods.<\/li>\n\n\n\n<li>Apply Heron&rsquo;s formula.<\/li>\n\n\n\n<li>Find circumference and use it to calculate distance travelled.<\/li>\n\n\n\n<li>Calculate the area of a circle and quadrant.<\/li>\n\n\n\n<li>Derive and apply the area formula of a trapezium.<\/li>\n\n\n\n<li>Find the area of a kite.<\/li>\n\n\n\n<li>Understand how area changes when dimensions are scaled.<\/li>\n\n\n\n<li>Solve shaded-area and circle-packing problems.<\/li>\n\n\n\n<li>Use geometric dissection to prove area relationships.<\/li>\n\n\n\n<li>Apply the Pythagorean theorem through semicircle areas.<\/li>\n\n\n\n<li>Solve advanced problems involving circles, triangles and rectangles.<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">3. Mind Map<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>MEASURING SPACE: PERIMETER &amp; AREA<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&rarr; <strong>Triangle<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Area = &frac12; &times; base &times; height<\/li>\n\n\n\n<li>Heron&rsquo;s Formula<\/li>\n\n\n\n<li>Isosceles Triangle<\/li>\n\n\n\n<li>Right Triangle<\/li>\n<\/ul>\n\n\n\n<div class=\"internal-linking-related-contents\"><a href=\"https:\/\/mymockmate.com\/notes\/class-9-maths-exercise-1-1-solutions-coordinate-geometry-ncert-solutions\/\" class=\"template-2\"><span class=\"cta\">Related Topic to Read more<\/span><span class=\"postTitle\">NCERT Class 9 Maths Exercise 1.1 Solutions | Coordinate Geometry NCERT Solutions<\/span><\/a><\/div><p class=\"wp-block-paragraph\">&rarr; <strong>Circle<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Circumference = <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>&pi;<\/mi><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2\\pi r<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li>Area = <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\pi r^2<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li>Quadrant = <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\frac14\\pi r^2<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li>Semicircle = <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\frac12\\pi r^2<\/annotation><\/semantics><\/math><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">&rarr; <strong>Quadrilateral<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Rectangle<\/li>\n\n\n\n<li>Trapezium<\/li>\n\n\n\n<li>Kite<\/li>\n\n\n\n<li>Area relationships<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">&rarr; <strong>Scaling<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Length scale factor = <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li>Area scale factor = <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>k<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">k^2<\/annotation><\/semantics><\/math><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">&rarr; <strong>Area Models<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Algebraic identities<\/li>\n\n\n\n<li>Rearrangement<\/li>\n\n\n\n<li>Dissection<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">&rarr; <strong>Advanced Geometry<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Shaded regions<\/li>\n\n\n\n<li>Circle packing<\/li>\n\n\n\n<li>Congruent shapes<\/li>\n\n\n\n<li>Semicircle constructions<\/li>\n\n\n\n<li>Area proofs<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The chapter summary confirms the core circle, triangle, arc and sector formulas used throughout the chapter.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">4. Questions &amp; Complete Solutions<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Question 1<\/h2>\n\n\n\n<div class=\"internal-linking-related-contents\"><a href=\"https:\/\/mymockmate.com\/notes\/class-9-maths-exercise-1-2-solutions-coordinate-geometry-ncert-solutions\/\" class=\"template-2\"><span class=\"cta\">Related Topic to Read more<\/span><span class=\"postTitle\">NCERT Class 9 Maths Exercise 1.2 Solutions | Coordinate Geometry NCERT Solutions<\/span><\/a><\/div><p class=\"wp-block-paragraph\">The textbook first gives the identity:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(a+b)^2=a^2+2ab+b^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and asks students to draw figures representing:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>&minus;<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(a+b)(a-b)=a^2-b^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><mn>2<\/mn><mi>b<\/mi><mi>c<\/mi><mo>+<\/mo><mn>2<\/mn><mi>c<\/mi><mi>a<\/mi><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca.<\/annotation><\/semantics><\/math><\/p><div class=\"mymoc-middle mymoc-entity-placement\" id=\"mymoc-3118869982\"><div id=\"mymoc-3691946778\"><a href=\"https:\/\/amzn.to\/4xnw9oY\" aria-label=\"head-phones\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones.png 1301w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones-300x80.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones-1024x274.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones-768x205.png 768w\" sizes=\"(max-width: 1301px) 100vw, 1301px\" width=\"1301\" height=\"348\"><\/a><\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<h3 class=\"wp-block-heading\">(a) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>&minus;<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(a+b)(a-b)=a^2-b^2<\/annotation><\/semantics><\/math><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Start with a square of side <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Its area is:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">a^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Remove a smaller square of side <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math>:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">b^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<div class=\"internal-linking-related-contents\"><a href=\"https:\/\/mymockmate.com\/notes\/class-9-maths-end-of-chapter-exercise-solutions-coordinate-geometry-ncert-solutions\/\" class=\"template-2\"><span class=\"cta\">Related Topic to Read more<\/span><span class=\"postTitle\">NCERT Class 9 Maths End of Chapter Exercise Solutions | Coordinate Geometry<\/span><\/a><\/div><p class=\"wp-block-paragraph\">Therefore, remaining area is:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">a^2-b^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The remaining L-shaped region can be rearranged into a rectangle having dimensions:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>&times;<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>&minus;<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(a+b)\\times(a-b)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>&minus;<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{(a+b)(a-b)=a^2-b^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(b) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(a+b+c)^2<\/annotation><\/semantics><\/math><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Draw a square of side:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a+b+c<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Divide each side into lengths <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo separator=\"true\">,<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a,b,c<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The large square consists of:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">a^2<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">b^2<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">c^2<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li>two <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ab<\/annotation><\/semantics><\/math> rectangles<\/li>\n\n\n\n<li>two <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>b<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">bc<\/annotation><\/semantics><\/math> rectangles<\/li>\n\n\n\n<li>two <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>c<\/mi><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ca<\/annotation><\/semantics><\/math> rectangles<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><mn>2<\/mn><mi>b<\/mi><mi>c<\/mi><mo>+<\/mo><mn>2<\/mn><mi>c<\/mi><mi>a<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong> Both identities can be demonstrated by dividing a large square into smaller squares and rectangles.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 2<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">An isosceles triangle has perimeter <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>40<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">40<\/annotation><\/semantics><\/math> cm and equal sides <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>15<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">15<\/annotation><\/semantics><\/math> cm each. Find its area.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Base:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>40<\/mn><mo>&minus;<\/mo><mn>15<\/mn><mo>&minus;<\/mo><mn>15<\/mn><mo>=<\/mo><mn>10<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">40-15-15=10\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The altitude bisects the base:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>10<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>5<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{10}{2}=5\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using Pythagoras:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>h<\/mi><mo>=<\/mo><msqrt><mrow><msup><mn>15<\/mn><mn>2<\/mn><\/msup><mo>&minus;<\/mo><msup><mn>5<\/mn><mn>2<\/mn><\/msup><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">h=\\sqrt{15^2-5^2}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>225<\/mn><mo>&minus;<\/mo><mn>25<\/mn><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{225-25}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mn>200<\/mn><\/msqrt><mo>=<\/mo><mn>10<\/mn><msqrt><mn>2<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{200}=10\\sqrt2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>&times;<\/mo><mn>10<\/mn><mo>&times;<\/mo><mn>10<\/mn><msqrt><mn>2<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">A=\\frac12\\times10\\times10\\sqrt2<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>=<\/mo><mn>50<\/mn><msqrt><mn>2<\/mn><\/msqrt><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A=50\\sqrt2\\text{ cm}^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>50<\/mn><msqrt><mn>2<\/mn><\/msqrt><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><mo>&asymp;<\/mo><mn>70.71<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{50\\sqrt2\\text{ cm}^2}\\approx70.71\\text{ cm}^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 3<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">An isosceles triangle has base <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">10<\/annotation><\/semantics><\/math> cm and area <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>60<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">60\\text{ cm}^2<\/annotation><\/semantics><\/math>. Find the equal sides.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>60<\/mn><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>&times;<\/mo><mn>10<\/mn><mo>&times;<\/mo><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">60=\\frac12\\times10\\times h<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>h<\/mi><mo>=<\/mo><mn>12<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">h=12\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The altitude divides the base into two parts of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5<\/annotation><\/semantics><\/math> cm each.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Equal side:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><msqrt><mrow><msup><mn>12<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>5<\/mn><mn>2<\/mn><\/msup><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">s=\\sqrt{12^2+5^2}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>144<\/mn><mo>+<\/mo><mn>25<\/mn><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{144+25}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mn>169<\/mn><\/msqrt><mo>=<\/mo><mn>13<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{169}=13<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>13<\/mn><mtext>&nbsp;cm&nbsp;and&nbsp;<\/mtext><mn>13<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{13\\text{ cm and }13\\text{ cm}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 4<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A right-angled triangle has area <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>54<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">54\\text{ cm}^2<\/annotation><\/semantics><\/math>. One leg is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">12<\/annotation><\/semantics><\/math> cm. Find its perimeter.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let the other leg be <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>.<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>54<\/mn><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>&times;<\/mo><mn>12<\/mn><mo>&times;<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">54=\\frac12\\times12\\times x<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>9<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">x=9\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hypotenuse:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>c<\/mi><mo>=<\/mo><msqrt><mrow><msup><mn>12<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>9<\/mn><mn>2<\/mn><\/msup><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">c=\\sqrt{12^2+9^2}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>144<\/mn><mo>+<\/mo><mn>81<\/mn><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mn>225<\/mn><\/msqrt><mo>=<\/mo><mn>15<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{144+81} =\\sqrt{225}=15<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Perimeter:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>12<\/mn><mo>+<\/mo><mn>9<\/mn><mo>+<\/mo><mn>15<\/mn><mo>=<\/mo><mn>36<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">12+9+15=36<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>36<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{36\\text{ cm}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 5<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The sides of a triangle are in the ratio <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mo>:<\/mo><mn>3<\/mn><mo>:<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2:3:4<\/annotation><\/semantics><\/math>, and its perimeter is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>45<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">45<\/annotation><\/semantics><\/math> cm. Find its area.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Total ratio:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2+3+4=9<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">One part:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>45<\/mn><mo>&divide;<\/mo><mn>9<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">45\\div9=5<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, sides are:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>10<\/mn><mo separator=\"true\">,<\/mo><mtext>&nbsp;<\/mtext><mn>15<\/mn><mo separator=\"true\">,<\/mo><mtext>&nbsp;<\/mtext><mn>20<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">10,\\ 15,\\ 20\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Semi-perimeter:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mrow><mn>10<\/mn><mo>+<\/mo><mn>15<\/mn><mo>+<\/mo><mn>20<\/mn><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>22.5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">s=\\frac{10+15+20}{2}=22.5<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By Heron&rsquo;s formula:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><msqrt><mrow><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><mi>s<\/mi><mo>&minus;<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>s<\/mi><mo>&minus;<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>s<\/mi><mo>&minus;<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">A=\\sqrt{s(s-a)(s-b)(s-c)}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>22.5<\/mn><mo stretchy=\"false\">(<\/mo><mn>12.5<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>7.5<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>2.5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{22.5(12.5)(7.5)(2.5)}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mfrac><mrow><mn>75<\/mn><msqrt><mn>15<\/mn><\/msqrt><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac{75\\sqrt{15}}4<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mn>75<\/mn><msqrt><mn>15<\/mn><\/msqrt><\/mrow><mn>4<\/mn><\/mfrac><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{75\\sqrt{15}}4\\text{ cm}^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Approximately,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>72.62<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{72.62\\text{ cm}^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 6<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The sides of a triangle are <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">7<\/annotation><\/semantics><\/math> cm, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">24<\/annotation><\/semantics><\/math> cm and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>25<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">25<\/annotation><\/semantics><\/math> cm. Find its area in two different ways.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Method 1: Right Triangle<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Since<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mn>7<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>24<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mn>25<\/mn><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">7^2+24^2=25^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">the triangle is right-angled.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>&times;<\/mo><mn>7<\/mn><mo>&times;<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">A=\\frac12\\times7\\times24<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>=<\/mo><mn>84<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A=84\\text{ cm}^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Method 2: Heron&rsquo;s Formula<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mrow><mn>7<\/mn><mo>+<\/mo><mn>24<\/mn><mo>+<\/mo><mn>25<\/mn><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>28<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">s=\\frac{7+24+25}{2}=28<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><msqrt><mrow><mn>28<\/mn><mo stretchy=\"false\">(<\/mo><mn>28<\/mn><mo>&minus;<\/mo><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>28<\/mn><mo>&minus;<\/mo><mn>24<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>28<\/mn><mo>&minus;<\/mo><mn>25<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">A=\\sqrt{28(28-7)(28-24)(28-25)}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>28<\/mn><mo>&times;<\/mo><mn>21<\/mn><mo>&times;<\/mo><mn>4<\/mn><mo>&times;<\/mo><mn>3<\/mn><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{28\\times21\\times4\\times3}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mn>7056<\/mn><\/msqrt><mo>=<\/mo><mn>84<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{7056}=84<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>84<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{84\\text{ cm}^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Both methods give the same answer.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 7<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A bicycle wheel has diameter <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>60<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">60<\/annotation><\/semantics><\/math> cm. Find the distance travelled after 100 rotations.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Circumference:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>C<\/mi><mo>=<\/mo><mi>&pi;<\/mi><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">C=\\pi d<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><mo>&times;<\/mo><mn>60<\/mn><mo>=<\/mo><mfrac><mn>1320<\/mn><mn>7<\/mn><\/mfrac><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac{22}{7}\\times60 =\\frac{1320}{7}\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For 100 rotations:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mn>100<\/mn><mo>&times;<\/mo><mfrac><mn>1320<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">D=100\\times\\frac{1320}{7}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mfrac><mn>132000<\/mn><mn>7<\/mn><\/mfrac><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac{132000}{7}\\text{ cm}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>&asymp;<\/mo><mn>18857.14<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\approx18857.14\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Converting into metres:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>188.57<\/mn><mtext>&nbsp;m&nbsp;approximately<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{188.57\\text{ m approximately}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 8<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Find the area of a quadrant whose circumference is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>66<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">66<\/annotation><\/semantics><\/math> cm.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><mi>&pi;<\/mi><mi>r<\/mi><mo>=<\/mo><mn>66<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2\\pi r=66<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>&pi;<\/mi><mo>=<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\pi=\\frac{22}{7}<\/annotation><\/semantics><\/math>:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><mo>&times;<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><mi>r<\/mi><mo>=<\/mo><mn>66<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2\\times\\frac{22}{7}r=66<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>r<\/mi><mo>=<\/mo><mn>10.5<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">r=10.5\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Quadrant area:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">A=\\frac14\\pi r^2<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mo>&times;<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><mo>&times;<\/mo><mo stretchy=\"false\">(<\/mo><mn>10.5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac14\\times\\frac{22}{7}\\times(10.5)^2<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>86.625<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">=\\boxed{86.625\\text{ cm}^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 9<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A car wheel has an outer radius of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>28<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">28<\/annotation><\/semantics><\/math> cm. Find:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Distance travelled in one complete turn.<\/li>\n\n\n\n<li>Number of turns in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">1<\/annotation><\/semantics><\/math> km.<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Circumference:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>C<\/mi><mo>=<\/mo><mn>2<\/mn><mi>&pi;<\/mi><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">C=2\\pi r<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mn>2<\/mn><mo>&times;<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><mo>&times;<\/mo><mn>28<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">=2\\times\\frac{22}{7}\\times28<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mn>176<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">=176\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Distance in one turn<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>176<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{176\\text{ cm}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Turns in 1 km<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>1<\/mn><mtext>&nbsp;km<\/mtext><mo>=<\/mo><mn>100000<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">1\\text{ km}=100000\\text{ cm}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>N<\/mi><mo>=<\/mo><mfrac><mn>100000<\/mn><mn>176<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">N=\\frac{100000}{176}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>&asymp;<\/mo><mn>568.18<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\approx568.18<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>176<\/mn><mtext>&nbsp;cm&nbsp;per&nbsp;turn<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{176\\text{ cm per turn}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and approximately<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>568.18<\/mn><mtext>&nbsp;turns<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{568.18\\text{ turns}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 10<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Two rectangles have the same area and the same perimeter. Are they necessarily congruent?<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let the sides be <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mi>a<\/mi><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A=ab<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Perimeter:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>P<\/mi><mo>=<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">P=2(a+b)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If both area and perimeter are fixed, then both:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a+b<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>a<\/mi><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ab<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">are fixed.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math> are determined as the two roots of the same quadratic equation:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>x<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2-(a+b)x+ab=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence the two rectangles have the same pair of side lengths.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mtext>Yes,&nbsp;the&nbsp;rectangles&nbsp;are&nbsp;congruent.<\/mtext><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Yes, the rectangles are congruent.}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">They may differ only in orientation.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Questions 11&ndash;15<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Question 11<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Show that the area of a trapezium is:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac12(a+b)h<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo separator=\"true\">,<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a,b<\/annotation><\/semantics><\/math> are the parallel sides.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Take two identical copies of the trapezium.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">When joined suitably, they form a parallelogram.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The parallelogram has:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>base<\/mtext><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\text{base}=a+b<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and height <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">h<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>A<\/mi><mtext>parallelogram<\/mtext><\/msub><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A_{\\text{parallelogram}}=(a+b)h<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since it consists of two equal trapeziums:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><mi>A<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2A=(a+b)h<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A=\\frac12(a+b)h}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 12<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Show the trapezium formula by dividing the trapezium into two triangles.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Divide the trapezium into two triangles.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Their areas are:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>a<\/mi><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac12ah<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac12bh<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>a<\/mi><mi>h<\/mi><mo>+<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A=\\frac12ah+\\frac12bh<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A=\\frac12(a+b)h<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A=\\frac12(a+b)h}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 13<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Show how two identical trapeziums can form a parallelogram.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Take two congruent trapeziums and rotate one of them appropriately.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Join their non-parallel sides.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">They form a parallelogram whose:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>base = <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a+b<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li>height = <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">h<\/annotation><\/semantics><\/math><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Hence,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>A<\/mi><mtext>parallelogram<\/mtext><\/msub><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A_{\\text{parallelogram}}=(a+b)h<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since two trapeziums form it:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>A<\/mi><mtext>trapezium<\/mtext><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A_{\\text{trapezium}} =\\frac12(a+b)h<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 14<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Show that the area of a kite is half the product of its diagonals, using algebra and geometry.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Algebraic Method<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let the diagonals be <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>d<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">d_1<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>d<\/mi><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">d_2<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The diagonals of a kite are perpendicular.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">They divide the kite into four right triangles.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Total area:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>4<\/mn><mo>&times;<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>d<\/mi><mn>1<\/mn><\/msub><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>d<\/mi><mn>2<\/mn><\/msub><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">4\\times\\frac12 \\left(\\frac{d_1}{2}\\right) \\left(\\frac{d_2}{2}\\right)<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>d<\/mi><mn>1<\/mn><\/msub><msub><mi>d<\/mi><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac12d_1d_2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>d<\/mi><mn>1<\/mn><\/msub><msub><mi>d<\/mi><mn>2<\/mn><\/msub><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A=\\frac12d_1d_2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Geometrical Method<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Draw both diagonals.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">They divide the kite into four right-angled triangles. Combining their areas gives exactly half the product of the diagonals.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mtext>Area&nbsp;of&nbsp;kite<\/mtext><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>d<\/mi><mn>1<\/mn><\/msub><msub><mi>d<\/mi><mn>2<\/mn><\/msub><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Area of kite}=\\frac12d_1d_2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 15<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">(i) Rectangle<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If rectangle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABCD<\/annotation><\/semantics><\/math> has sides <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo separator=\"true\">,<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a,b<\/annotation><\/semantics><\/math>, its area is:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>a<\/mi><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ab<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Rectangle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mi>Q<\/mi><mi>R<\/mi><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PQRS<\/annotation><\/semantics><\/math> has sides <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>a<\/mi><mo separator=\"true\">,<\/mo><mn>2<\/mn><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2a,2b<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Its area:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>4<\/mn><mi>a<\/mi><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">(2a)(2b)=4ab<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>A<\/mi><mrow><mi>P<\/mi><mi>Q<\/mi><mi>R<\/mi><mi>S<\/mi><\/mrow><\/msub><mo>=<\/mo><mn>4<\/mn><msub><mi>A<\/mi><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><\/msub><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A_{PQRS}=4A_{ABCD}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Yes, four copies can be arranged to form the larger rectangle.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(ii) Triangle<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If every side is doubled, the linear scale factor is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area scale factor:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mn>2<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2^2=4<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>A<\/mi><mrow><mi>P<\/mi><mi>Q<\/mi><mi>R<\/mi><\/mrow><\/msub><mo>=<\/mo><mn>4<\/mn><msub><mi>A<\/mi><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><\/msub><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A_{PQR}=4A_{ABC}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Four copies can be arranged to form the larger similar triangle.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(iii) Triangle<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If every side is tripled:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Scale&nbsp;factor<\/mtext><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Scale factor}=3<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Area&nbsp;scale&nbsp;factor<\/mtext><mo>=<\/mo><msup><mn>3<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area scale factor}=3^2=9<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>A<\/mi><mrow><mi>P<\/mi><mi>Q<\/mi><mi>R<\/mi><\/mrow><\/msub><mo>=<\/mo><mn>9<\/mn><msub><mi>A<\/mi><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><\/msub><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A_{PQR}=9A_{ABC}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nine copies can be arranged to form the larger similar triangle.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 16<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find the fraction of the shaded area in:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(a) Fig. 6.43 &ndash; Triangle<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Using the equal-area divisions shown in the figure, the shaded central region is equal to half of the complete triangle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mtext>Shaded&nbsp;fraction<\/mtext><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Shaded fraction}=\\frac12}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The figure-based solution can be obtained by using the fact that a median divides a triangle into two equal areas.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(b) Fig. 6.44 &ndash; Square<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Draw lines parallel to the sides of the square through the vertices of the shaded region.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The square divides into <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>25<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">25<\/annotation><\/semantics><\/math> equal small regions, and the shaded portion corresponds to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5<\/annotation><\/semantics><\/math> of them.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Fraction<\/mtext><mo>=<\/mo><mfrac><mn>5<\/mn><mn>25<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Fraction}=\\frac5{25}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mfrac><mn>1<\/mn><mn>5<\/mn><\/mfrac><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac15}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 17<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find the fraction of the rectangle covered by the circles.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Fig. 6.45<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let radius of each circle be <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Rectangle dimensions:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>6<\/mn><mi>r<\/mi><mo>&times;<\/mo><mn>2<\/mn><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">6r\\times2r<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Rectangle area:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>12<\/mn><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">12r^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area of 3 circles:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>3<\/mn><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">3\\pi r^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Fraction:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mn>3<\/mn><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><mn>12<\/mn><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo>=<\/mo><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mfrac><mi>&pi;<\/mi><mn>4<\/mn><\/mfrac><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{3\\pi r^2}{12r^2} =\\boxed{\\frac{\\pi}{4}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Fig. 6.46<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Rectangle dimensions:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>8<\/mn><mi>r<\/mi><mo>&times;<\/mo><mn>2<\/mn><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">8r\\times2r<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>16<\/mn><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">16r^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Four circles:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>4<\/mn><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">4\\pi r^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Fraction:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mn>4<\/mn><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><mn>16<\/mn><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo>=<\/mo><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mfrac><mi>&pi;<\/mi><mn>4<\/mn><\/mfrac><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{4\\pi r^2}{16r^2} =\\boxed{\\frac{\\pi}{4}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">In both figures:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mi>&pi;<\/mi><mn>4<\/mn><\/mfrac><mo>&asymp;<\/mo><mn>78.5<\/mn><mi mathvariant=\"normal\">%<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{\\pi}{4}\\approx78.5\\%}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 18<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Make a conjecture about circles fitted into a rectangle and test it for 10, 20 and 50 circles.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conjecture<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math> equal circles of radius <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math> are arranged in one row inside a rectangle:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Rectangle&nbsp;length<\/mtext><mo>=<\/mo><mn>2<\/mn><mi>n<\/mi><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Rectangle length}=2nr<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Rectangle&nbsp;breadth<\/mtext><mo>=<\/mo><mn>2<\/mn><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Rectangle breadth}=2r<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>A<\/mi><mtext>rectangle<\/mtext><\/msub><mo>=<\/mo><mn>4<\/mn><mi>n<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">A_{\\text{rectangle}}=4nr^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math> circles:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>A<\/mi><mtext>circles<\/mtext><\/msub><mo>=<\/mo><mi>n<\/mi><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">A_{\\text{circles}}=n\\pi r^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><msub><mi>A<\/mi><mtext>circles<\/mtext><\/msub><msub><mi>A<\/mi><mtext>rectangle<\/mtext><\/msub><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>n<\/mi><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><mn>4<\/mn><mi>n<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{A_{\\text{circles}}}{A_{\\text{rectangle}}} = \\frac{n\\pi r^2}{4nr^2}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mfrac><mi>&pi;<\/mi><mn>4<\/mn><\/mfrac><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{\\pi}{4}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">For 10 circles<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mfrac><mi>&pi;<\/mi><mn>4<\/mn><\/mfrac><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{\\pi}{4}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">For 20 circles<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mfrac><mi>&pi;<\/mi><mn>4<\/mn><\/mfrac><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{\\pi}{4}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">For 50 circles<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mfrac><mi>&pi;<\/mi><mn>4<\/mn><\/mfrac><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{\\pi}{4}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The number of circles cancels out.<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mtext>Fraction&nbsp;covered<\/mtext><mo>=<\/mo><mfrac><mi>&pi;<\/mi><mn>4<\/mn><\/mfrac><mo>&asymp;<\/mo><mn>78.5<\/mn><mi mathvariant=\"normal\">%<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Fraction covered}=\\frac{\\pi}{4}\\approx78.5\\%}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 19<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Nine identical rectangles are arranged as shown. Their combined area is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>72<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">72\\text{ cm}^2<\/annotation><\/semantics><\/math>. Find the perimeter of each small rectangle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let the dimensions of each small rectangle be <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From the arrangement:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>4<\/mn><mi>b<\/mi><mo>=<\/mo><mn>5<\/mn><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">4b=5a<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mn>4<\/mn><mn>5<\/mn><\/mfrac><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a=\\frac45b<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The large rectangle has dimensions:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>4<\/mn><mi>b<\/mi><mo>&times;<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">4b\\times(a+b)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Its area is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>72<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">72<\/annotation><\/semantics><\/math>:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>4<\/mn><mi>b<\/mi><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>4<\/mn><mn>5<\/mn><\/mfrac><mi>b<\/mi><mo>+<\/mo><mi>b<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mn>72<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4b\\left(\\frac45b+b\\right)=72<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>4<\/mn><mi>b<\/mi><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>9<\/mn><mn>5<\/mn><\/mfrac><mi>b<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mn>72<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4b\\left(\\frac95b\\right)=72<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>36<\/mn><mn>5<\/mn><\/mfrac><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>72<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{36}{5}b^2=72<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b^2=10<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><msqrt><mn>10<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">b=\\sqrt{10}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mrow><mn>4<\/mn><msqrt><mn>10<\/mn><\/msqrt><\/mrow><mn>5<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a=\\frac{4\\sqrt{10}}5<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Perimeter:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>P<\/mi><mo>=<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">P=2(a+b)<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mn>2<\/mn><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mn>4<\/mn><msqrt><mn>10<\/mn><\/msqrt><\/mrow><mn>5<\/mn><\/mfrac><mo>+<\/mo><msqrt><mn>10<\/mn><\/msqrt><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">=2\\left(\\frac{4\\sqrt{10}}5+\\sqrt{10}\\right)<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mn>18<\/mn><msqrt><mn>10<\/mn><\/msqrt><\/mrow><mn>5<\/mn><\/mfrac><mtext>&nbsp;cm<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">=\\boxed{\\frac{18\\sqrt{10}}5\\text{ cm}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Approximately,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>11.38<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{11.38\\text{ cm}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 20<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Show that the blue and red triangles have equal areas and explain how the blue triangle can be rearranged to cover the red triangle.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The points on the opposite side divide it into three equal parts.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The blue and red triangles have:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>equal bases,<\/li>\n\n\n\n<li>the same altitude from the common vertex to the opposite side.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Since:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A=\\frac12bh<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and both <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">h<\/annotation><\/semantics><\/math> are equal,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>A<\/mi><mtext>blue<\/mtext><\/msub><mo>=<\/mo><msub><mi>A<\/mi><mtext>red<\/mtext><\/msub><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A_{\\text{blue}}=A_{\\text{red}}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Rearrangement<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The blue triangle can be divided into suitable smaller pieces along lines parallel to the base and the sides. These pieces can then be translated\/rearranged to occupy exactly the same area as the red triangle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, the equality is a consequence of <strong>equal base &times; equal height<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 21<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Show that shaded regions <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B<\/annotation><\/semantics><\/math> have equal areas.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let the side of the square be <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area of the quadrant:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\frac14\\pi r^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Each semicircle has diameter <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math>, so radius:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>r<\/mi><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac r2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area of one semicircle:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>&pi;<\/mi><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>r<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mrow><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>8<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac12\\pi\\left(\\frac r2\\right)^2 =\\frac{\\pi r^2}{8}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Two semicircles:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><mo>&times;<\/mo><mfrac><mrow><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>8<\/mn><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">2\\times\\frac{\\pi r^2}{8} =\\frac{\\pi r^2}{4}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mtext>Area&nbsp;of&nbsp;quadrant<\/mtext><mo>=<\/mo><mtext>sum&nbsp;of&nbsp;areas&nbsp;of&nbsp;two&nbsp;semicircles<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Area of quadrant}=\\text{sum of areas of two semicircles}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The common parts cancel when comparing the two shaded regions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Area}(A)=\\text{Area}(B)}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 22<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Four semicircles form a four-petalled flower inside a square of side <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2<\/annotation><\/semantics><\/math> units. Find its perimeter and area.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Each semicircle has diameter <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2<\/annotation><\/semantics><\/math>, so:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>r<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">r=1<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Perimeter<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The boundary consists of eight quarter-circle arcs of radius <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">1<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">One quarter-circle arc:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>&pi;<\/mi><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mi>&pi;<\/mi><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac14(2\\pi r)=\\frac{\\pi}{2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Eight arcs:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>8<\/mn><mo>&times;<\/mo><mfrac><mi>&pi;<\/mi><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">8\\times\\frac{\\pi}{2}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>P<\/mi><mo>=<\/mo><mn>4<\/mn><mi>&pi;<\/mi><mtext>&nbsp;units<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{P=4\\pi\\text{ units}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Area<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Each petal is formed from two <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>90<\/mn><mo>&#8728;<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90^\\circ<\/annotation><\/semantics><\/math> sectors and two right triangles.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area of two sectors:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><mo>&times;<\/mo><mfrac><mi>&pi;<\/mi><mn>4<\/mn><\/mfrac><mo>=<\/mo><mfrac><mi>&pi;<\/mi><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">2\\times\\frac{\\pi}{4}=\\frac{\\pi}{2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area of two right triangles:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><mo>&times;<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2\\times\\frac12=1<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area of one petal:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>&pi;<\/mi><mn>2<\/mn><\/mfrac><mo>&minus;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{\\pi}{2}-1<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There are four petals:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mn>4<\/mn><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>&pi;<\/mi><mn>2<\/mn><\/mfrac><mo>&minus;<\/mo><mn>1<\/mn><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">A=4\\left(\\frac{\\pi}{2}-1\\right)<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>=<\/mo><mn>2<\/mn><mi>&pi;<\/mi><mo>&minus;<\/mo><mn>4<\/mn><mtext>&nbsp;square&nbsp;units<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A=2\\pi-4\\text{ square units}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 23<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Two concentric circles have a chord <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><mo>=<\/mo><mi>l<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC=l<\/annotation><\/semantics><\/math> of the larger circle touching the smaller circle at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math>. Show that the area between the circles is:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>&pi;<\/mi><msup><mi>l<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\frac14\\pi l^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let the outer radius be <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math> and inner radius be <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math> touches the smaller circle:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>O<\/mi><mi>A<\/mi><mo>&perp;<\/mo><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">OA\\perp BC<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>O<\/mi><mi>A<\/mi><mo>=<\/mo><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">OA=r<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Also,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>O<\/mi><mi>B<\/mi><mo>=<\/mo><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">OB=R<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Half the chord:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>=<\/mo><mfrac><mi>l<\/mi><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">AB=\\frac l2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By Pythagoras:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>l<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">R^2=r^2+\\left(\\frac l2\\right)^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><msup><mi>l<\/mi><mn>2<\/mn><\/msup><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">R^2-r^2=\\frac{l^2}{4}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area between circles:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>&pi;<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>R<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\pi(R^2-r^2)<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mfrac><mrow><mi>&pi;<\/mi><msup><mi>l<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>4<\/mn><\/mfrac><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">=\\boxed{\\frac{\\pi l^2}{4}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 24<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Show that:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area}(A)+\\text{Area}(B)=\\text{Area}(C)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">for semicircles constructed on the sides of a right-angled triangle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let the legs of the right triangle be <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo separator=\"true\">,<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a,b<\/annotation><\/semantics><\/math>, and hypotenuse be <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By Pythagoras:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">a^2+b^2=c^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area of semicircle on <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>&pi;<\/mi><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>a<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mrow><mi>&pi;<\/mi><msup><mi>a<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>8<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">A=\\frac12\\pi\\left(\\frac a2\\right)^2 =\\frac{\\pi a^2}{8}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Similarly,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>B<\/mi><mo>=<\/mo><mfrac><mrow><mi>&pi;<\/mi><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>8<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">B=\\frac{\\pi b^2}{8}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>C<\/mi><mo>=<\/mo><mfrac><mrow><mi>&pi;<\/mi><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>8<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">C=\\frac{\\pi c^2}{8}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>+<\/mo><mi>B<\/mi><mo>=<\/mo><mfrac><mrow><mi>&pi;<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><mn>8<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">A+B =\\frac{\\pi(a^2+b^2)}8<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">a^2+b^2=c^2<\/annotation><\/semantics><\/math>,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>+<\/mo><mi>B<\/mi><mo>=<\/mo><mfrac><mrow><mi>&pi;<\/mi><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>8<\/mn><\/mfrac><mo>=<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A+B=\\frac{\\pi c^2}{8}=C<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>+<\/mo><mi>B<\/mi><mo>=<\/mo><mi>C<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A+B=C}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is a beautiful area-based interpretation of the Pythagorean theorem.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 25<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Two congruent circles of radius <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math> pass through each other&rsquo;s centres. Find the area of their common region.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The distance between the centres is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The common chord subtends <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>60<\/mn><mo>&#8728;<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">60^\\circ<\/annotation><\/semantics><\/math> at each centre.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">One half of the common region consists of:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Sector&nbsp;<\/mtext><msup><mn>60<\/mn><mo>&#8728;<\/mo><\/msup><mo>&minus;<\/mo><mtext>Equilateral&nbsp;triangle<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Sector }60^\\circ-\\text{Equilateral triangle}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sector area:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>60<\/mn><mn>360<\/mn><\/mfrac><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mrow><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{60}{360}\\pi r^2 =\\frac{\\pi r^2}{6}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Equilateral triangle area:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><msqrt><mn>3<\/mn><\/msqrt><mn>4<\/mn><\/mfrac><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{\\sqrt3}{4}r^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">One half:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>6<\/mn><\/mfrac><mo>&minus;<\/mo><mfrac><msqrt><mn>3<\/mn><\/msqrt><mn>4<\/mn><\/mfrac><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{\\pi r^2}{6}-\\frac{\\sqrt3}{4}r^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There are two equal halves.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mn>2<\/mn><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>6<\/mn><\/mfrac><mo>&minus;<\/mo><mfrac><mrow><msqrt><mn>3<\/mn><\/msqrt><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>4<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">A=2\\left(\\frac{\\pi r^2}{6}-\\frac{\\sqrt3r^2}{4}\\right)<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>&pi;<\/mi><mn>3<\/mn><\/mfrac><mo>&minus;<\/mo><mfrac><msqrt><mn>3<\/mn><\/msqrt><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{ A=\\left(\\frac{\\pi}{3}-\\frac{\\sqrt3}{2}\\right)r^2 }<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">or<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>&pi;<\/mi><mo>&minus;<\/mo><mn>3<\/mn><msqrt><mn>3<\/mn><\/msqrt><\/mrow><mn>6<\/mn><\/mfrac><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{ A=\\frac{2\\pi-3\\sqrt3}{6}r^2 }<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 26<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">In Fig. 6.54, three triangles have areas <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mo separator=\"true\">,<\/mo><mi>B<\/mi><mo separator=\"true\">,<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A,B,C<\/annotation><\/semantics><\/math>. Show that the area of the rectangle is:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mfrac><mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo>+<\/mo><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>B<\/mi><mo>+<\/mo><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mi>C<\/mi><\/mfrac><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{2(A+C)(B+C)}{C}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Proof<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let the dimensions associated with the figure be <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo separator=\"true\">,<\/mo><mi>c<\/mi><mo separator=\"true\">,<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a,b,c,d<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The rectangle has dimensions:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mtext>and<\/mtext><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(a+b)\\quad\\text{and}\\quad(c+d)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>A<\/mi><mtext>rectangle<\/mtext><\/msub><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">A_{\\text{rectangle}}=(a+b)(c+d)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From the figure:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A=\\frac12ac<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>B<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B=\\frac12bd<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>C<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">C=\\frac12bc<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>+<\/mo><mi>C<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>a<\/mi><mi>c<\/mi><mo>+<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A+C=\\frac12ac+\\frac12bc<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>c<\/mi><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac12c(a+b)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Similarly,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>B<\/mi><mo>+<\/mo><mi>C<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>d<\/mi><mo>+<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B+C=\\frac12bd+\\frac12bc<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mo stretchy=\"false\">(<\/mo><mi>d<\/mi><mo>+<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac12b(d+c)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo>+<\/mo><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>B<\/mi><mo>+<\/mo><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mi>C<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2(A+C)(B+C)}C<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mrow><mo fence=\"true\">[<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>c<\/mi><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"true\">]<\/mo><\/mrow><mrow><mo fence=\"true\">[<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"true\">]<\/mo><\/mrow><\/mrow><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>c<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">= \\frac{ 2\\left[\\frac12c(a+b)\\right] \\left[\\frac12b(c+d)\\right] }{ \\frac12bc }<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">After cancellation:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">=(a+b)(c+d)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But this is exactly the area of the rectangle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mtext>Area&nbsp;of&nbsp;rectangle<\/mtext><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo>+<\/mo><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>B<\/mi><mo>+<\/mo><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mi>C<\/mi><\/mfrac><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{ \\text{Area of rectangle} = \\frac{2(A+C)(B+C)}{C} }<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The same algebraic relationship is independently reflected in published solution treatments of the figure.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 27<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Show that the two shaded regions formed by a quarter circle, a semicircle and a triangle have equal areas.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let the right triangle have equal perpendicular sides <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then its hypotenuse is:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>r<\/mi><msqrt><mn>2<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">r\\sqrt2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Area of quarter circle<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>A<\/mi><mi>Q<\/mi><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">A_Q=\\frac14\\pi r^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Area of semicircle<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Its diameter is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><msqrt><mn>2<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">r\\sqrt2<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore radius:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>r<\/mi><msqrt><mn>2<\/mn><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{r\\sqrt2}{2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>A<\/mi><mi>S<\/mi><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>&pi;<\/mi><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>r<\/mi><msqrt><mn>2<\/mn><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">A_S = \\frac12\\pi \\left(\\frac{r\\sqrt2}{2}\\right)^2<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mfrac><mrow><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac{\\pi r^2}{4}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>A<\/mi><mi>Q<\/mi><\/msub><mo>=<\/mo><msub><mi>A<\/mi><mi>S<\/mi><\/msub><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A_Q=A_S}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The triangle is common to the two constructions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, after subtracting the same triangular area from equal circular areas, the two remaining shaded regions are equal.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mtext>Area&nbsp;of&nbsp;first&nbsp;shaded&nbsp;region<\/mtext><mo>=<\/mo><mtext>Area&nbsp;of&nbsp;second&nbsp;shaded&nbsp;region<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Area of first shaded region} = \\text{Area of second shaded region}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">5. Common Errors<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Error 1: Using the wrong radius<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If diameter is given:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>r<\/mi><mo>=<\/mo><mfrac><mi>d<\/mi><mn>2<\/mn><\/mfrac><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{r=\\frac d2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Do not use the diameter directly in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">A=\\pi r^2<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Error 2: Forgetting the factor <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac12<\/annotation><\/semantics><\/math><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For a triangle:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>h<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A=\\frac12bh}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Error 3: Confusing circumference and area<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>C<\/mi><mo>=<\/mo><mn>2<\/mn><mi>&pi;<\/mi><mi>r<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{C=2\\pi r}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">but<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>=<\/mo><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A=\\pi r^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Error 4: Wrong scaling rule<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If lengths are multiplied by <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math>, areas are multiplied by:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><msup><mi>k<\/mi><mn>2<\/mn><\/msup><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{k^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">not <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Error 5: Using Heron&rsquo;s formula without semi-perimeter<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Always calculate:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{s=\\frac{a+b+c}{2}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">before applying Heron&rsquo;s formula.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Error 6: Forgetting that <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>&pi;<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\pi<\/annotation><\/semantics><\/math> is an approximation<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The textbook explicitly notes that:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>&pi;<\/mi><mo mathvariant=\"normal\">&ne;<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\pi\\ne\\frac{22}{7}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">but<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>&pi;<\/mi><mo>&asymp;<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\pi\\approx\\frac{22}{7}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Error 7: Adding shaded areas directly<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">In complex figures, identify:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Required region = Total region &minus; unwanted region<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">or use:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Required region = larger region &minus; common region.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Error 8: Assuming equal area means congruent<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Two figures may have the same area without having the same shape.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">6. PYQs \/ Exam-Oriented Practice<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Note:<\/strong> The uploaded PDF does not label any questions as previous-year questions (PYQs). Therefore, the following are best presented on MyMockMate as <strong>PYQ-style \/ exam-oriented practice questions<\/strong>, rather than claiming they are official previous-year questions.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">PYQ-Style Question 1<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">An isosceles triangle has equal sides <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>13<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">13<\/annotation><\/semantics><\/math> cm and base <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">10<\/annotation><\/semantics><\/math> cm. Find its area.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>60<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{60\\text{ cm}^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">PYQ-Style Question 2<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A wheel of radius <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>21<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">21<\/annotation><\/semantics><\/math> cm makes 100 complete rotations. Find the distance travelled.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>100<\/mn><mo>&times;<\/mo><mn>2<\/mn><mo>&times;<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><mo>&times;<\/mo><mn>21<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">100\\times2\\times\\frac{22}{7}\\times21<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>13200<\/mn><mtext>&nbsp;cm<\/mtext><mo>=<\/mo><mn>132<\/mn><mtext>&nbsp;m<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{13200\\text{ cm}=132\\text{ m}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">PYQ-Style Question 3<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The parallel sides of a trapezium are <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>15<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">15<\/annotation><\/semantics><\/math> cm and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>25<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">25<\/annotation><\/semantics><\/math> cm and its height is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">8<\/annotation><\/semantics><\/math> cm. Find its area.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mo>+<\/mo><mn>25<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>8<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>160<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\frac12(15+25)(8) =\\boxed{160\\text{ cm}^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">PYQ-Style Question 4<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A square has side <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">14<\/annotation><\/semantics><\/math> cm. Find the area of its inscribed circle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Radius:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>r<\/mi><mo>=<\/mo><mn>7<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">r=7\\text{ cm}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><mo>&times;<\/mo><mn>49<\/mn><mo>=<\/mo><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>154<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">A=\\frac{22}{7}\\times49 =\\boxed{154\\text{ cm}^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">PYQ-Style Question 5<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The dimensions of a rectangle are doubled. How does its area change?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mtext>Area&nbsp;becomes&nbsp;4&nbsp;times<\/mtext><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Area becomes 4 times}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">PYQ-Style Question 6<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Show that the areas of semicircles constructed on the three sides of a right triangle satisfy the same relationship as the squares of its sides.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This follows from:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">a^2+b^2=c^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>A<\/mi><mtext>semicircle<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><mi>&pi;<\/mi><msup><mi>d<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>8<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">A_{\\text{semicircle}}=\\frac{\\pi d^2}{8}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>+<\/mo><mi>B<\/mi><mo>=<\/mo><mi>C<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A+B=C}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">7. FAQ<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Q1. What is the most important formula for a circle?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>=<\/mo><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A=\\pi r^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Q2. What is the formula for circumference?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>C<\/mi><mo>=<\/mo><mn>2<\/mn><mi>&pi;<\/mi><mi>r<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{C=2\\pi r}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Q3. What is the area of a quadrant?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac14\\pi r^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Q4. When should Heron&rsquo;s formula be used?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Use Heron&rsquo;s formula when the <strong>three sides of a triangle are known<\/strong>.<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>=<\/mo><msqrt><mrow><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><mi>s<\/mi><mo>&minus;<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>s<\/mi><mo>&minus;<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>s<\/mi><mo>&minus;<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A=\\sqrt{s(s-a)(s-b)(s-c)}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Q5. What happens to area when all dimensions are doubled?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Area becomes:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>4<\/mn><mtext>&nbsp;times<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{4\\text{ times}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Q6. What happens when all dimensions are tripled?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Area becomes:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>9<\/mn><mtext>&nbsp;times<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{9\\text{ times}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Q7. Why is the fraction of circles covering the rectangle always <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>&pi;<\/mi><mi mathvariant=\"normal\">\/<\/mi><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\pi\/4<\/annotation><\/semantics><\/math>?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Because:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>n<\/mi><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mi>&pi;<\/mi><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{n\\pi r^2}{(2nr)(2r)} =\\frac{\\pi}{4}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The number <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math> and radius <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math> cancel.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Q8. What is the trapezium area formula?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac12(a+b)h}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Q9. What is the kite area formula?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>d<\/mi><mn>1<\/mn><\/msub><msub><mi>d<\/mi><mn>2<\/mn><\/msub><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac12d_1d_2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Q10. What is the key idea behind Question 24?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The areas of semicircles are proportional to the <strong>squares of their diameters<\/strong>, so Pythagoras gives:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mo>+<\/mo><mi>B<\/mi><mo>=<\/mo><mi>C<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A+B=C}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">8. Summary<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">The End-of-Chapter Exercises consolidate the complete chapter through numerical problems, proofs, constructions and area puzzles.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Must-Remember Formulas<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>A<\/mi><mi mathvariant=\"normal\">&#9651;<\/mi><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>h<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A_{\\triangle}=\\frac12bh}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>A<\/mi><mtext>circle<\/mtext><\/msub><mo>=<\/mo><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A_{\\text{circle}}=\\pi r^2}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>C<\/mi><mtext>circle<\/mtext><\/msub><mo>=<\/mo><mn>2<\/mn><mi>&pi;<\/mi><mi>r<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{C_{\\text{circle}}=2\\pi r}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>A<\/mi><mtext>quadrant<\/mtext><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>&pi;<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A_{\\text{quadrant}}=\\frac14\\pi r^2}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>A<\/mi><mtext>trapezium<\/mtext><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A_{\\text{trapezium}}=\\frac12(a+b)h}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>A<\/mi><mtext>kite<\/mtext><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>d<\/mi><mn>1<\/mn><\/msub><msub><mi>d<\/mi><mn>2<\/mn><\/msub><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A_{\\text{kite}}=\\frac12d_1d_2}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>A<\/mi><mrow><mi mathvariant=\"normal\">&#9651;<\/mi><mo separator=\"true\">,<\/mo><mtext>Heron<\/mtext><\/mrow><\/msub><mo>=<\/mo><msqrt><mrow><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><mi>s<\/mi><mo>&minus;<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>s<\/mi><mo>&minus;<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>s<\/mi><mo>&minus;<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{ A_{\\triangle,\\text{Heron}} = \\sqrt{s(s-a)(s-b)(s-c)} }<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Key Concepts<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Area models<\/strong> can prove algebraic identities.<\/li>\n\n\n\n<li><strong>Heron&rsquo;s formula<\/strong> is useful when all three sides are known.<\/li>\n\n\n\n<li><strong>Scaling lengths by kk<\/strong> scales areas by <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>k<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">k^2<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Circle area<\/strong> depends on <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">r^2<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Semicircle areas<\/strong> can demonstrate the Pythagorean theorem.<\/li>\n\n\n\n<li>Complex figures can often be solved by <strong>breaking them into simpler shapes<\/strong>.<\/li>\n\n\n\n<li>Equal bases and equal heights lead to equal triangle areas.<\/li>\n\n\n\n<li>Circle-packing problems often reveal a constant ratio of:<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mfrac><mi>&pi;<\/mi><mn>4<\/mn><\/mfrac><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{\\pi}{4}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The chapter&rsquo;s own summary reinforces the central formulas for circumference, arc length, triangle area, Heron&rsquo;s formula, circle area and sector area.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Takeaway<\/h3>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"wp-block-paragraph\"><strong>Measure smartly: break complicated figures into simple shapes, apply the right formula, and look for relationships rather than calculating blindly.<\/strong><\/p>\n<\/blockquote>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>MyMockMate Learning Tip:<\/strong><br><strong>Understand &rarr; Visualise &rarr; Apply Formula &rarr; Simplify &rarr; Verify &rarr; Practise<\/strong>.<\/p>\n<div class=\"mymoc-bottom mymoc-entity-placement\" id=\"mymoc-938941267\"><div id=\"mymoc-1845666043\"><a href=\"https:\/\/amzn.to\/3S4CntS\" aria-label=\"Stationeries\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/Stationeries.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/Stationeries.png 1303w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/Stationeries-300x89.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/Stationeries-1024x304.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/Stationeries-768x228.png 768w\" sizes=\"(max-width: 1303px) 100vw, 1303px\" width=\"1080\" height=\"100\"><\/a><\/div><\/div>","protected":false},"excerpt":{"rendered":"<p>End-of-Chapter Exercises &ndash; Complete Solutions 1. Intro This chapter develops the idea that area and perimeter can be understood through decomposition, rearrangement, similarity, scaling and&#8230;<\/p>\n","protected":false},"author":1,"featured_media":6892,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"postBodyCss":"","postBodyMargin":[],"postBodyPadding":[],"postBodyBackground":{"backgroundType":"classic","gradient":""},"footnotes":""},"categories":[4,11],"tags":[],"class_list":["post-6891","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-9","category-maths-class-9"],"featured_image_src":"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/09\/Class-9-Maths-Chapter-6-\u2013-Measuring-Space-Perimeter-and-Area.png","author_info":{"display_name":"Team Mymockmate","author_link":"https:\/\/mymockmate.com\/notes\/author\/bsm_adm\/"},"_links":{"self":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/6891","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/comments?post=6891"}],"version-history":[{"count":1,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/6891\/revisions"}],"predecessor-version":[{"id":6893,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/6891\/revisions\/6893"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media\/6892"}],"wp:attachment":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media?parent=6891"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/categories?post=6891"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/tags?post=6891"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}