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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.

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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

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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

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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 $AB=d$,$$r=\frac{d}{2}$$

The smallest circle occurs when $AB$AB is the diameter.

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Exercise 5.1 Solutions

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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

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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

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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

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Question 4

What is the least possible radius of a circle through two points A and B?

Solution

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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.

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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:$$r=\frac{AB}{2}$$r=2AB​

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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

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Q2.

The circumcentre of a right triangle lies:

A. Inside
B. Outside
C. At midpoint of hypotenuse
D. At centroid

Answer: C

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Q3.

The smallest circle through points A and B has radius:

A. AB

B. $2AB$2AB

C. $\frac{AB}{2}$2AB​

D. $\frac{AB}{4}$4AB​

Answer: C

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Q4.

The circumcentre is obtained by intersection of:

A. Medians
B. Altitudes
C. Angle bisectors
D. Perpendicular bisectors

Answer: D

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Frequently Asked Questions (FAQs)

Q1. What is a circumcircle?

A circle passing through all three vertices of a triangle.

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Q2. What is the circumcentre?

The intersection point of the perpendicular bisectors of the sides of a triangle.

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Q3. Can the circumcentre lie outside a triangle?

Yes, in an obtuse-angled triangle.

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Q4. Can three collinear points have a circumcircle?

No.

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Q5. Why are OA, OB and OC equal?

Because they are radii of the same circle.

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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

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