{"id":6877,"date":"2026-09-12T15:09:25","date_gmt":"2026-09-12T09:39:25","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=6877"},"modified":"2026-09-12T15:10:05","modified_gmt":"2026-09-12T09:40:05","slug":"class-9-maths-exercise-6-2-complete-solutions","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/class-9-maths-exercise-6-2-complete-solutions\/","title":{"rendered":"Class 9 | Maths | Exercise 6.2 \u2014 Complete Solutions"},"content":{"rendered":"<div class=\"mymoc-top mymoc-entity-placement\" id=\"mymoc-1777632811\"><div id=\"mymoc-2009225822\"><a href=\"https:\/\/amzn.to\/4ehJfM4\" aria-label=\"students acessories\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/students-acessories.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/students-acessories.png 1301w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/students-acessories-300x85.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/students-acessories-1024x290.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/students-acessories-768x218.png 768w\" sizes=\"(max-width: 1301px) 100vw, 1301px\" width=\"1080\" height=\"100\"><\/a><\/div><\/div>\n<h2 class=\"wp-block-heading\">Chapter 6: Measuring Space &mdash; Perimeter and Area<\/h2>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">1. Quick Revision<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Area of a Triangle<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The area of a triangle 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\"><mrow><mtext>Area<\/mtext><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>&times;<\/mo><mtext>base<\/mtext><mo>&times;<\/mo><mtext>height<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Area}=\\frac12\\times\\text{base}\\times\\text{height}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">When the three sides of a triangle are known, we can use <strong>Heron&rsquo;s Formula<\/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><mtext>Area<\/mtext><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{\\text{Area}=\\sqrt{s(s-a)(s-b)(s-c)}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where<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\">Here, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>s<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">s<\/annotation><\/semantics><\/math> is the <strong>semiperimeter<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Area of a Trapezium<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If the parallel sides are <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>, and the height is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">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><mtext>Area<\/mtext><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{\\text{Area}=\\frac12(a+b)h}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Area of a Rhombus<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If its diagonals are <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>:<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>=<\/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}=\\frac12d_1d_2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Important Area Principle<\/h3>\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\">Triangles having the <strong>same base and the same height<\/strong> have equal areas.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Also, triangles having equal bases and lying between the same parallel lines have equal areas.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">2. Important Facts<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Perimeter is the total length around a closed figure.<\/li>\n\n\n\n<li>Area measures the region enclosed by a figure.<\/li>\n\n\n\n<li>The area of a triangle is half the area of a parallelogram having the same base and height.<\/li>\n\n\n\n<li>Heron&rsquo;s formula is useful when all three sides of a triangle are known.<\/li>\n\n\n\n<li>In a parallelogram, triangles formed on the same base and between the same parallels have equal areas.<\/li>\n\n\n\n<li>A median divides a triangle into two triangles of equal area.<\/li>\n\n\n\n<li>The diagonals of a rhombus bisect each other at right angles.<\/li>\n\n\n\n<li>The area of a rhombus is half the product of its diagonals.<\/li>\n\n\n\n<li>When the side lengths of a triangular plot are given in a ratio, first find the actual side lengths using the perimeter.<\/li>\n\n\n\n<li>In geometry proofs, comparing <strong>common bases, heights, parallel lines and midpoints<\/strong> is often the key to proving equal areas.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The chapter introduces Heron&rsquo;s formula for finding the area of a triangle from its three sides.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">3. Image<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Area Relationships in Exercise 6.2<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A useful concept diagram for this exercise is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Triangle &rarr; Trapezium &rarr; Rhombus &rarr; Parallelogram &rarr; Median &rarr; Quadrilateral<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Key formulas:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">&#9651;<\/mi><mo>:<\/mo><mspace width=\"1em\"><\/mspace><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\">\\triangle:\\quad A=\\frac12 bh<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Trapezium<\/mtext><mo>:<\/mo><mspace width=\"1em\"><\/mspace><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\">\\text{Trapezium}:\\quad A=\\frac12(a+b)h<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Rhombus<\/mtext><mo>:<\/mo><mspace width=\"1em\"><\/mspace><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><annotation encoding=\"application\/x-tex\">\\text{Rhombus}:\\quad A=\\frac12d_1d_2<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Triangle&nbsp;by&nbsp;Heron&rsquo;s&nbsp;Formula<\/mtext><mo>:<\/mo><mspace width=\"1em\"><\/mspace><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\">\\text{Triangle by Heron&rsquo;s Formula}:\\quad A=\\sqrt{s(s-a)(s-b)(s-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\">4. Questions<\/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\"><strong>Find the area of triangle ADE in Fig. 6.31.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">From the figure:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">DC=10<\/annotation><\/semantics><\/math> cm<\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">BC=8<\/annotation><\/semantics><\/math> cm<\/li>\n\n\n\n<li>Since <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> is a rectangle, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>D<\/mi><mo>=<\/mo><mi>B<\/mi><mi>C<\/mi><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">AD=BC=8<\/annotation><\/semantics><\/math> cm.<\/li>\n\n\n\n<li>The perpendicular distance from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">E<\/annotation><\/semantics><\/math> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math> is <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.<\/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><mtext>Area&nbsp;of&nbsp;<\/mtext><mi mathvariant=\"normal\">&#9651;<\/mi><mi>A<\/mi><mi>D<\/mi><mi>E<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>&times;<\/mo><mi>A<\/mi><mi>D<\/mi><mo>&times;<\/mo><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area of }\\triangle ADE =\\frac12\\times AD\\times 10<\/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><mo>&times;<\/mo><mn>8<\/mn><mo>&times;<\/mo><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac12\\times8\\times10<\/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>40<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">=\\boxed{40\\text{ cm}^2}<\/annotation><\/semantics><\/math><\/p><div class=\"mymoc-middle mymoc-entity-placement\" id=\"mymoc-1867105755\"><div id=\"mymoc-1924634200\"><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>\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>40<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{40\\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 2<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The parallel sides of a trapezium are 40 cm and 20 cm. Its non-parallel sides are equal, each being 26 cm. Find its area.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The difference between the parallel sides is:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>40<\/mn><mo>&minus;<\/mo><mn>20<\/mn><mo>=<\/mo><mn>20<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">40-20=20\\text{ cm}<\/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\">Since the trapezium is isosceles, the extra length is divided equally between the two sides.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, horizontal projection on each side is:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>20<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>10<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{20}{2}=10\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using Pythagoras&rsquo; theorem:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>h<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>10<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mn>26<\/mn><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">h^2+10^2=26^2<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>h<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>676<\/mn><mo>&minus;<\/mo><mn>100<\/mn><mo>=<\/mo><mn>576<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">h^2=676-100=576<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>h<\/mi><mo>=<\/mo><mn>24<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">h=24\\text{ cm}<\/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><mtext>Area<\/mtext><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mn>40<\/mn><mo>+<\/mo><mn>20<\/mn><mo stretchy=\"false\">)<\/mo><mo>&times;<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area} =\\frac12(40+20)\\times24<\/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><mo>&times;<\/mo><mn>60<\/mn><mo>&times;<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac12\\times60\\times24<\/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>720<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">=\\boxed{720\\text{ cm}^2}<\/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>720<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{720\\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\"><strong>Find the area of a triangle whose two sides are 8 cm and 11 cm, and whose perimeter is 32 cm.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let the third side 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>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Given perimeter:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>8<\/mn><mo>+<\/mo><mn>11<\/mn><mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>32<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">8+11+x=32<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>13<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">x=13\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, the three sides are:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>8<\/mn><mo separator=\"true\">,<\/mo><mtext>&nbsp;<\/mtext><mn>11<\/mn><mo separator=\"true\">,<\/mo><mtext>&nbsp;<\/mtext><mn>13<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">8,\\ 11,\\ 13<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Semiperimeter:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mrow><mn>8<\/mn><mo>+<\/mo><mn>11<\/mn><mo>+<\/mo><mn>13<\/mn><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">s=\\frac{8+11+13}{2}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mn>16<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">s=16\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using 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>16<\/mn><mo stretchy=\"false\">(<\/mo><mn>16<\/mn><mo>&minus;<\/mo><mn>8<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>16<\/mn><mo>&minus;<\/mo><mn>11<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>16<\/mn><mo>&minus;<\/mo><mn>13<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{16(16-8)(16-11)(16-13)}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>16<\/mn><mo>&times;<\/mo><mn>8<\/mn><mo>&times;<\/mo><mn>5<\/mn><mo>&times;<\/mo><mn>3<\/mn><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{16\\times8\\times5\\times3}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mn>1920<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{1920}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mn>8<\/mn><msqrt><mn>30<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=8\\sqrt{30}<\/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><mn>8<\/mn><msqrt><mn>30<\/mn><\/msqrt><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A=8\\sqrt{30}\\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><mi>A<\/mi><mo>&asymp;<\/mo><mn>43.82<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">A\\approx43.82\\text{ cm}^2<\/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>8<\/mn><msqrt><mn>30<\/mn><\/msqrt><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><mo>&asymp;<\/mo><mn>43.82<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{8\\sqrt{30}\\text{ cm}^2\\approx43.82\\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 4<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The sides of a triangular plot are in the ratio 3:5:73:5:7. Its perimeter is 300 m. Find its area.<\/strong><\/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\" display=\"block\"><semantics><mrow><mn>3<\/mn><mi>x<\/mi><mo separator=\"true\">,<\/mo><mtext>&nbsp;<\/mtext><mn>5<\/mn><mi>x<\/mi><mo separator=\"true\">,<\/mo><mtext>&nbsp;<\/mtext><mn>7<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">3x,\\ 5x,\\ 7x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Their sum is 300 m:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mi>x<\/mi><mo>=<\/mo><mn>300<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x+5x+7x=300<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>15<\/mn><mi>x<\/mi><mo>=<\/mo><mn>300<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">15x=300<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>20<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x=20<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the sides are:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>60<\/mn><mtext>&nbsp;m<\/mtext><mo separator=\"true\">,<\/mo><mtext>&nbsp;<\/mtext><mn>100<\/mn><mtext>&nbsp;m<\/mtext><mo separator=\"true\">,<\/mo><mtext>&nbsp;<\/mtext><mn>140<\/mn><mtext>&nbsp;m<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">60\\text{ m},\\ 100\\text{ m},\\ 140\\text{ m}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Semiperimeter:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mrow><mn>60<\/mn><mo>+<\/mo><mn>100<\/mn><mo>+<\/mo><mn>140<\/mn><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>150<\/mn><mtext>&nbsp;m<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">s=\\frac{60+100+140}{2}=150\\text{ m}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using Heron&rsquo;s formula:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><msqrt><mrow><mn>150<\/mn><mo stretchy=\"false\">(<\/mo><mn>150<\/mn><mo>&minus;<\/mo><mn>60<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>150<\/mn><mo>&minus;<\/mo><mn>100<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>150<\/mn><mo>&minus;<\/mo><mn>140<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">A=\\sqrt{150(150-60)(150-100)(150-140)}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>150<\/mn><mo>&times;<\/mo><mn>90<\/mn><mo>&times;<\/mo><mn>50<\/mn><mo>&times;<\/mo><mn>10<\/mn><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{150\\times90\\times50\\times10}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>6<\/mn><mo separator=\"true\">,<\/mo><mn>750<\/mn><mo separator=\"true\">,<\/mo><mn>000<\/mn><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{6,750,000}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mn>1500<\/mn><msqrt><mn>3<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=1500\\sqrt3<\/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><mn>1500<\/mn><msqrt><mn>3<\/mn><\/msqrt><msup><mtext>&nbsp;m<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A=1500\\sqrt3\\text{ m}^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><mi>A<\/mi><mo>&asymp;<\/mo><mn>2598.08<\/mn><msup><mtext>&nbsp;m<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{A\\approx2598.08\\text{ m}^2}<\/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>1500<\/mn><msqrt><mn>3<\/mn><\/msqrt><msup><mtext>&nbsp;m<\/mtext><mn>2<\/mn><\/msup><mo>&asymp;<\/mo><mn>2598.08<\/mn><msup><mtext>&nbsp;m<\/mtext><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{1500\\sqrt3\\text{ m}^2\\approx2598.08\\text{ m}^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 5<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>One diagonal of a rhombus is twice as long as the other. If its area is 128&nbsp;cm, find the length of the shorter diagonal.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let the shorter diagonal 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> cm.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then the longer diagonal is:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">2x\\text{ cm}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Area of a rhombus:<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><msub><mi>d<\/mi><mn>1<\/mn><\/msub><msub><mi>d<\/mi><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">A=\\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><mn>128<\/mn><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">128=\\frac12(x)(2x)<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>128<\/mn><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">128=x^2<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><msqrt><mn>128<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">x=\\sqrt{128}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>8<\/mn><msqrt><mn>2<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">x=8\\sqrt2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence, the shorter diagonal 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\"><mrow><mn>8<\/mn><msqrt><mn>2<\/mn><\/msqrt><mtext>&nbsp;cm<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{8\\sqrt2\\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>11.31<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{11.31\\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 6<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>ABCD is a parallelogram. P and Q are any two points on side AB. What can you say about the ratio<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">)<\/mo><mo>:<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>Q<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">?<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area}(\\triangle PCD):\\text{Area}(\\triangle QCD)?<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Triangles <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PCD<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>Q<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">QCD<\/annotation><\/semantics><\/math> have the <strong>same base CD<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q<\/annotation><\/semantics><\/math> lie on <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>, and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>&#8741;<\/mo><mi>C<\/mi><mi>D<\/mi><mo separator=\"true\">,<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">AB\\parallel CD,<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">the perpendicular distance from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q<\/annotation><\/semantics><\/math> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CD<\/annotation><\/semantics><\/math> is the same.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, both triangles have:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>the same base <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CD<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li>the same height.<\/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><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>Q<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area}(\\triangle PCD) = \\text{Area}(\\triangle QCD)<\/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><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">)<\/mo><mo>:<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>Q<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><mo>:<\/mo><mn>1<\/mn><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Area}(\\triangle PCD):\\text{Area}(\\triangle QCD)=1:1}<\/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>1<\/mn><mo>:<\/mo><mn>1<\/mn><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{1:1}<\/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 7<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>O is any point on the diagonal PR of parallelogram PQRS. Prove that the areas of triangles PSO and PQO are equal.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Proof<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Consider <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>S<\/mi><mi>O<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle PSO<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>Q<\/mi><mi>O<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle PQO<\/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>O<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">O<\/annotation><\/semantics><\/math> lies on diagonal <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PR<\/annotation><\/semantics><\/math>, both triangles have their bases <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mi>O<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PO<\/annotation><\/semantics><\/math> on the same straight line <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PR<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now, in parallelogram <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>,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>P<\/mi><mi>S<\/mi><mo>&#8741;<\/mo><mi>Q<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PS\\parallel QR<\/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>P<\/mi><mi>Q<\/mi><mo>&#8741;<\/mo><mi>S<\/mi><mi>R<\/mi><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PQ\\parallel SR.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The points <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">S<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q<\/annotation><\/semantics><\/math> are equally distant from the diagonal <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PR<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, the two triangles have:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>the same base <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mi>O<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PO<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li>equal perpendicular heights.<\/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><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>S<\/mi><mi>O<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>Q<\/mi><mi>O<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area}(\\triangle PSO) = \\text{Area}(\\triangle PQO)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence proved:<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 mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>S<\/mi><mi>O<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>Q<\/mi><mi>O<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Area}(\\triangle PSO)=\\text{Area}(\\triangle PQO)}<\/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 8<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>If the midpoints of the sides of a 4-gon are joined in order, prove that the area of the parallelogram formed is half the area of the given 4-gon.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Proof<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let <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> be the given quadrilateral.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mo separator=\"true\">,<\/mo><mi>Q<\/mi><mo separator=\"true\">,<\/mo><mi>R<\/mi><mo separator=\"true\">,<\/mo><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P,Q,R,S<\/annotation><\/semantics><\/math> be the midpoints of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo separator=\"true\">,<\/mo><mi>B<\/mi><mi>C<\/mi><mo separator=\"true\">,<\/mo><mi>C<\/mi><mi>D<\/mi><mo separator=\"true\">,<\/mo><mi>D<\/mi><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB,BC,CD,DA<\/annotation><\/semantics><\/math>, respectively.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Joining <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mo separator=\"true\">,<\/mo><mi>Q<\/mi><mo separator=\"true\">,<\/mo><mi>R<\/mi><mo separator=\"true\">,<\/mo><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P,Q,R,S<\/annotation><\/semantics><\/math> forms a parallelogram.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Because <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">S<\/annotation><\/semantics><\/math> are midpoints of <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> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math>, the segment <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PS<\/annotation><\/semantics><\/math> is parallel to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD<\/annotation><\/semantics><\/math> and has half its length.<\/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>Q<\/mi><mi>R<\/mi><mo>&#8741;<\/mo><mi>B<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">QR\\parallel BD<\/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>Q<\/mi><mi>R<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">QR=\\frac12BD.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Likewise,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>P<\/mi><mi>Q<\/mi><mo>&#8741;<\/mo><mi>A<\/mi><mi>C<\/mi><mo separator=\"true\">,<\/mo><mspace width=\"2em\"><\/mspace><mi>P<\/mi><mi>Q<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>A<\/mi><mi>C<\/mi><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PQ\\parallel AC,\\qquad PQ=\\frac12AC.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, <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> is a parallelogram.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The four corner triangles outside <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> together have an area equal to half the area of <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>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence the remaining central parallelogram has:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi>P<\/mi><mi>Q<\/mi><mi>R<\/mi><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">)<\/mo><mo>&minus;<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area}(PQRS) = \\text{Area}(ABCD)-\\frac12\\text{Area}(ABCD)<\/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><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi>P<\/mi><mi>Q<\/mi><mi>R<\/mi><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Area}(PQRS)=\\frac12\\text{Area}(ABCD)}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Hence proved.<\/h3>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 9<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>In &#9651;ABC\\triangle ABC, D is the midpoint of BC. Median AD is drawn. P is any point on AD. Show that<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>A<\/mi><mi>B<\/mi><mi>P<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>A<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area}(\\triangle ABP)=\\text{Area}(\\triangle ACP).<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Proof<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math> is the midpoint of <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>,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mo>=<\/mo><mi>D<\/mi><mi>C<\/mi><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD=DC.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Also, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math> lies on <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Consider triangles <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABP<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>C<\/mi><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ACP<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Take <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AP<\/annotation><\/semantics><\/math> as their common base.<\/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><mo separator=\"true\">,<\/mo><mi>D<\/mi><mo separator=\"true\">,<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B,D,C<\/annotation><\/semantics><\/math> are collinear and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math> is the midpoint of <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>, the perpendicular distances of <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>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">C<\/annotation><\/semantics><\/math> from line <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AP<\/annotation><\/semantics><\/math> are equal.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the two triangles have:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>equal bases <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AP<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li>equal heights.<\/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><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>A<\/mi><mi>B<\/mi><mi>P<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>A<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area}(\\triangle ABP) = \\text{Area}(\\triangle ACP).<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Hence proved.<\/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><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>A<\/mi><mi>B<\/mi><mi>P<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>A<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Area}(\\triangle ABP)=\\text{Area}(\\triangle ACP)}<\/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 10<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Given a square ABCD, P is a point inside it. PA, PB, PC and PD are joined. Find the ratio of the areas of the red region (&#9651;PAB+&#9651;PCD)(\\triangle PAB+\\triangle PCD) and the green region (&#9651;PBC+&#9651;PDA)(\\triangle PBC+\\triangle PDA).<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\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>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The four triangles formed are:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>A<\/mi><mi>B<\/mi><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>B<\/mi><mi>C<\/mi><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>C<\/mi><mi>D<\/mi><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>D<\/mi><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle PAB,\\quad \\triangle PBC,\\quad \\triangle PCD,\\quad \\triangle PDA<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The red region consists of:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>A<\/mi><mi>B<\/mi><mo>+<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle PAB+\\triangle PCD<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The green region consists of:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>B<\/mi><mi>C<\/mi><mo>+<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>D<\/mi><mi>A<\/mi><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle PBC+\\triangle PDA.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Notice that:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>A<\/mi><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area}(\\triangle PAB)+\\text{Area}(\\triangle PCD)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is equal to half the area of the square.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Similarly,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>B<\/mi><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>P<\/mi><mi>D<\/mi><mi>A<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area}(\\triangle PBC)+\\text{Area}(\\triangle PDA)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is also equal to half the area of the square.<\/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>Red&nbsp;area<\/mtext><mo>=<\/mo><mtext>Green&nbsp;area<\/mtext><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Red area}=\\text{Green area}.<\/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><mtext>Red&nbsp;area<\/mtext><mo>:<\/mo><mtext>Green&nbsp;area<\/mtext><mo>=<\/mo><mn>1<\/mn><mo>:<\/mo><mn>1<\/mn><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Red area}:\\text{Green area}=1:1}<\/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>1<\/mn><mo>:<\/mo><mn>1<\/mn><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{1:1}<\/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 11<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>In &#9651;ABC, D is the midpoint of AB. P is any point on BC, and Q is a point on AB such that CQ&#8741;PD. Prove that<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>B<\/mi><mi>P<\/mi><mi>Q<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area}(\\triangle BPQ) = \\frac12\\text{Area}(\\triangle ABC).<\/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:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mo>=<\/mo><mi>D<\/mi><mi>A<\/mi><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD=DA.<\/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>C<\/mi><mi>Q<\/mi><mo>&#8741;<\/mo><mi>P<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CQ\\parallel PD<\/annotation><\/semantics><\/math>, triangles formed by these parallel lines give a proportional relationship.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For a simple coordinate demonstration, take:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>B<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>A<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>C<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo separator=\"true\">,<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B=(0,0),\\quad A=(0,2),\\quad C=(c,0).<\/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>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math> is the midpoint of <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>,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D=(0,1).<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>P<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>p<\/mi><mo separator=\"true\">,<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P=(p,0).<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The slope of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PD<\/annotation><\/semantics><\/math> is:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mn>0<\/mn><mo>&minus;<\/mo><mn>1<\/mn><\/mrow><mrow><mi>p<\/mi><mo>&minus;<\/mo><mn>0<\/mn><\/mrow><\/mfrac><mo>=<\/mo><mo>&minus;<\/mo><mfrac><mn>1<\/mn><mi>p<\/mi><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{0-1}{p-0}=-\\frac1p.<\/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>C<\/mi><mi>Q<\/mi><mo>&#8741;<\/mo><mi>P<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CQ\\parallel PD<\/annotation><\/semantics><\/math>, the line through <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">C<\/annotation><\/semantics><\/math> parallel to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PD<\/annotation><\/semantics><\/math> meets <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> at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This gives:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>Q<\/mi><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mfrac><mi>c<\/mi><mi>p<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q=\\left(0,\\frac cp\\right).<\/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>B<\/mi><mi>Q<\/mi><mo>=<\/mo><mfrac><mi>c<\/mi><mi>p<\/mi><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BQ=\\frac cp.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>P<\/mi><mo>=<\/mo><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BP=p<\/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><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>B<\/mi><mi>P<\/mi><mi>Q<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>&times;<\/mo><mi>B<\/mi><mi>P<\/mi><mo>&times;<\/mo><mi>B<\/mi><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area}(\\triangle BPQ) = \\frac12\\times BP\\times BQ<\/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><mo>&times;<\/mo><mi>p<\/mi><mo>&times;<\/mo><mfrac><mi>c<\/mi><mi>p<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac12\\times p\\times\\frac cp<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mfrac><mi>c<\/mi><mn>2<\/mn><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac c2.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The area of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">&#9651;<\/mi><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle ABC<\/annotation><\/semantics><\/math> is:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>&times;<\/mo><mi>B<\/mi><mi>C<\/mi><mo>&times;<\/mo><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac12\\times BC\\times AB<\/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><mo>&times;<\/mo><mi>c<\/mi><mo>&times;<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">=\\frac12\\times c\\times2<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mi>c<\/mi><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">=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><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>B<\/mi><mi>P<\/mi><mi>Q<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area}(\\triangle BPQ) = \\frac12\\text{Area}(\\triangle ABC).<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence proved:<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 mathvariant=\"normal\">&#9651;<\/mi><mi>B<\/mi><mi>P<\/mi><mi>Q<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mtext>Area<\/mtext><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">&#9651;<\/mi><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Area}(\\triangle BPQ)=\\frac12\\text{Area}(\\triangle ABC)}<\/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. HOTS &mdash; Higher Order Thinking Skills<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">HOTS Question 1<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A triangle has sides 10 cm, 17 cm and 21 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\"><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>17<\/mn><mo>+<\/mo><mn>21<\/mn><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">s=\\frac{10+17+21}{2}=24<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using Heron&rsquo;s formula:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><msqrt><mrow><mn>24<\/mn><mo stretchy=\"false\">(<\/mo><mn>24<\/mn><mo>&minus;<\/mo><mn>10<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>24<\/mn><mo>&minus;<\/mo><mn>17<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>24<\/mn><mo>&minus;<\/mo><mn>21<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">A=\\sqrt{24(24-10)(24-17)(24-21)}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>24<\/mn><mo>&times;<\/mo><mn>14<\/mn><mo>&times;<\/mo><mn>7<\/mn><mo>&times;<\/mo><mn>3<\/mn><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{24\\times14\\times7\\times3}<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><msqrt><mn>7056<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">=\\sqrt{7056}<\/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>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<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">HOTS Question 2<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A rhombus has area <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>180<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">180\\text{ cm}^2<\/annotation><\/semantics><\/math>. One diagonal is 12 cm. Find the other diagonal.<\/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><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><annotation encoding=\"application\/x-tex\">A=\\frac12d_1d_2<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>180<\/mn><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>&times;<\/mo><mn>12<\/mn><mo>&times;<\/mo><msub><mi>d<\/mi><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">180=\\frac12\\times12\\times d_2<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>180<\/mn><mo>=<\/mo><mn>6<\/mn><msub><mi>d<\/mi><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">180=6d_2<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>d<\/mi><mn>2<\/mn><\/msub><mo>=<\/mo><mn>30<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">d_2=30<\/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>30<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{30\\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\">HOTS Question 3<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Two triangles have the same base and the same area. What can you conclude about their heights?<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\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><mo separator=\"true\">,<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">A=\\frac12bh,<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and both triangles have the same <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>, their heights must also be 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\"><mtext>Their&nbsp;heights&nbsp;are&nbsp;equal.<\/mtext><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Their heights are equal.}}<\/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\">6. Worksheet<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">A. Fill in the blanks<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>The area of a triangle is ________.<\/li>\n\n\n\n<li>Heron&rsquo;s formula uses the ________ of a triangle.<\/li>\n\n\n\n<li>The area of a rhombus is ________.<\/li>\n\n\n\n<li>A median divides a triangle into two ________ triangles.<\/li>\n\n\n\n<li>The area of a trapezium is ________.<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">B. Solve<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Find the area of a triangle whose sides are 13 cm, 14 cm and 15 cm.<\/li>\n\n\n\n<li>Find the area of a trapezium with parallel sides 18 cm and 12 cm and height 7 cm.<\/li>\n\n\n\n<li>A rhombus has diagonals 16 cm and 10 cm. Find its area.<\/li>\n\n\n\n<li>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 54 cm. Find its area.<\/li>\n\n\n\n<li>A triangle and a parallelogram have the same base and height. What is the ratio of their areas?<\/li>\n\n\n\n<li>A rhombus has area <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>96<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">96\\text{ cm}^2<\/annotation><\/semantics><\/math> and one diagonal is 12 cm. Find the other diagonal.<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Answer Key<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>84<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">84\\text{ cm}^2<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>105<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">105\\text{ cm}^2<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>80<\/mn><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">80\\text{ cm}^2<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>27<\/mn><mo>:<\/mo><mn>81<\/mn><mo stretchy=\"false\">?<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">27:81?<\/annotation><\/semantics><\/math> &mdash; <strong>Use Heron&rsquo;s formula after finding the sides 12,18,2412,18,24<\/strong>; area <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>=<\/mo><mn>54<\/mn><msqrt><mn>7<\/mn><\/msqrt><msup><mtext>&nbsp;cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">=54\\sqrt7\\text{ cm}^2<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1<\/mn><mo>:<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">1:2<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>16<\/mn><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">16\\text{ cm}<\/annotation><\/semantics><\/math><\/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\">7. FAQ<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Q1. When should I use Heron&rsquo;s formula?<\/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> but its height is not directly available.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Q2. What is the semiperimeter?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The semiperimeter is half the perimeter:<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<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Q3. Why is the area of a triangle half of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>b<\/mi><mi>a<\/mi><mi>s<\/mi><mi>e<\/mi><mo>&times;<\/mo><mi>h<\/mi><mi>e<\/mi><mi>i<\/mi><mi>g<\/mi><mi>h<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">base\\times height<\/annotation><\/semantics><\/math>?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A triangle can be paired with an identical triangle to form a parallelogram. The triangle occupies half of that parallelogram.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Q4. Why do triangles on the same base sometimes have equal areas?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If their third vertices lie on a line parallel to the base, their heights are equal. 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><mo separator=\"true\">,<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">A=\\frac12bh,<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">their areas are equal.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Q5. Why is the area of a rhombus <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><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><annotation encoding=\"application\/x-tex\">\\frac12d_1d_2<\/annotation><\/semantics><\/math>?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The diagonals divide the rhombus into four right-angled triangles. Adding their areas 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><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<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Q6. Does the position of P affect the answer in Question 10?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">No. As long as <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math> remains inside the square, the combined areas of the opposite pairs remain equal. Therefore, the ratio is always:<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>1<\/mn><mo>:<\/mo><mn>1<\/mn><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{1:1}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Q7. What is the most important idea in Exercise 6.2?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The key idea is to <strong>compare bases and heights<\/strong>. Many of the proof-based questions can be solved without calculating actual numerical areas.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Exam Tip<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">For Exercise 6.2, remember these four formulas:<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>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>rhombus<\/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{rhombus}}=\\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><mi mathvariant=\"normal\">&#9651;<\/mi><\/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}=\\sqrt{s(s-a)(s-b)(s-c)}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">These formulas, together with the concepts of <strong>equal bases, equal heights, parallel lines and midpoints<\/strong>, cover almost the entire exercise. The source PDF&rsquo;s Exercise 6.2 specifically includes all 11 questions addressed above.<\/p>\n<div class=\"mymoc-bottom mymoc-entity-placement\" id=\"mymoc-4103805366\"><div id=\"mymoc-1525289827\"><a href=\"https:\/\/amzn.to\/4fDWGbw\" aria-label=\"smartwatch\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smartwatch.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smartwatch.png 1717w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smartwatch-300x71.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smartwatch-1024x243.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smartwatch-768x182.png 768w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smartwatch-1536x364.png 1536w\" sizes=\"(max-width: 1717px) 100vw, 1717px\" width=\"1717\" height=\"407\"><\/a><\/div><\/div>","protected":false},"excerpt":{"rendered":"<p>Chapter 6: Measuring Space &mdash; Perimeter and Area 1. Quick Revision Area of a Triangle The area of a triangle is:Area=12&times;base&times;height\\boxed{\\text{Area}=\\frac12\\times\\text{base}\\times\\text{height}} When the three sides&#8230;<\/p>\n","protected":false},"author":1,"featured_media":6878,"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-6877","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-Exercise-6.2-\u2014-Complete-Solutions.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\/6877","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=6877"}],"version-history":[{"count":1,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/6877\/revisions"}],"predecessor-version":[{"id":6879,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/6877\/revisions\/6879"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media\/6878"}],"wp:attachment":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media?parent=6877"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/categories?post=6877"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/tags?post=6877"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}