NCERT Class 9 Maths Exercise 5.1 Solutions | Circles
Short Introduction
A circle is one of the most symmetrical and fascinating geometric shapes. In this exercise, students learn how to construct the circumcircle of a triangle and understand the concept of the circumcentre. The exercise also explores circles passing through two or three points and the minimum radius of such circles.
Quick Information Box
| Chapter | Circles |
|---|---|
| Exercise | 5.1 |
| Class | 9 |
| Topic | Circumcircle and Circumcentre |
| Concepts Used | Perpendicular Bisector, Triangle Construction |
| Difficulty Level | Easy to Moderate |
Concepts Used (Topics Covered)
- Circle and its properties
- Circumcentre of a triangle
- Circumcircle of a triangle
- Perpendicular bisector
- Acute, obtuse and isosceles triangles
- Radius of a circle through two points
Important Formulas & Facts
1. Circumcentre
The point where the perpendicular bisectors of the sides of a triangle intersect.
2. Circumcircle
The circle passing through all three vertices of a triangle.
3. Radius of Smallest Circle Through Two Points
If ,
The smallest circle occurs when AB is the diameter.
Exercise 5.1 Solutions
Question 1
Draw ΔABC with AB = 5 cm, ∠A = 70° and ∠B = 60°. Draw the circumcircle of ΔABC. Is the centre inside or outside the triangle?
Solution
Question 2
Draw ΔABC with AB = 5 cm, ∠A = 100° and AC = 4 cm. Draw the circumcircle of ΔABC. Is the centre inside or outside the triangle?
Solution
Question 3
Draw ΔABC with AB = 6 cm, BC = 7 cm and CA = 7 cm. Draw the circumcircle of ΔABC. Let the circumcentre be O. Measure OA, OB and OC.
Solution
Question 4
What is the least possible radius of a circle through two points A and B?
Solution
Common Mistakes Students Make
❌ Drawing only one perpendicular bisector.
❌ Taking an arbitrary point as the centre.
❌ Forgetting that circumcentre may lie outside the triangle.
❌ Measuring unequal radii due to inaccurate construction.
❌ Confusing circumcentre with centroid.
Exam Tips
✅ Remember:
- Acute triangle → circumcentre inside.
- Right triangle → circumcentre at midpoint of hypotenuse.
- Obtuse triangle → circumcentre outside.
✅ The perpendicular bisectors always meet at one point.
✅ Radius of the smallest circle through two points:r=2AB
Practice MCQs
Q1.
The circumcentre of an acute triangle lies:
A. Outside the triangle
B. On a side
C. Inside the triangle
D. At a vertex
Answer: C
Q2.
The circumcentre of a right triangle lies:
A. Inside
B. Outside
C. At midpoint of hypotenuse
D. At centroid
Answer: C
Q3.
The smallest circle through points A and B has radius:
A. AB
B. 2AB
C. 2AB
D. 4AB
Answer: C
Q4.
The circumcentre is obtained by intersection of:
A. Medians
B. Altitudes
C. Angle bisectors
D. Perpendicular bisectors
Answer: D
Frequently Asked Questions (FAQs)
Q1. What is a circumcircle?
A circle passing through all three vertices of a triangle.
Q2. What is the circumcentre?
The intersection point of the perpendicular bisectors of the sides of a triangle.
Q3. Can the circumcentre lie outside a triangle?
Yes, in an obtuse-angled triangle.
Q4. Can three collinear points have a circumcircle?
No.
Q5. Why are OA, OB and OC equal?
Because they are radii of the same circle.
Final Answer Summary
| Question | Answer |
|---|---|
| Q1 | Circumcentre lies inside the triangle |
| Q2 | Circumcentre lies outside the triangle |
| Q3 | OA = OB = OC |
| Q4 | Least radius = AB/2 |
Call To Action (CTA)
📚 Loved these solutions?
Explore more NCERT Class 9 Maths Chapter-wise Solutions, Notes, MCQs, and Mock Tests on:
🚀 Learn Smart. Practice More. Score Better with MyMockMate!