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Madhyamik Mathematics Chapter 11 – Triangle Circumcircle Theorems | Questions and Answers

15 Sept 2026 0 views

Theorems Related to the Circumcircle of a Triangle is an important geometry topic for WBBSE Class 10 Mathematics. This chapter deals with the circumcircle of a triangle, its centre and radius, the circumcentre, the incenter, perpendicular bisectors of chords and the construction of circumcircles.

The practice material below has been organised directly from the supplied Madhyamik Mathematics Chapter 11 source into a cleaner, student-friendly format.

Triangle Circumcircle Theorems – Chapter 11

Core Concept: The circumcircle of a triangle is the circle passing through all three vertices of the triangle. Its centre is called the circumcentre and its radius is called the circumradius.

Important Terms for Quick Revision

Term Meaning
Incentre The centre of the incircle of a triangle.
Inradius The radius of the incircle.
Circumcentre The centre of the circumcircle of a triangle.
Circumradius The radius of the circumcircle.
Perpendicular Bisector The perpendicular bisectors of two chords intersect at the centre of the circle.

Fill in the Blanks

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1. The centre of the incircle of a triangle is called the ______ and its radius is called the ______.
Answer: Incentre, Inradius
2. The centre of the circumcircle of a triangle is called the ______ and its radius is called the ______.
Answer: Circumcentre, Circumradius
3. The point of intersection of the perpendicular bisectors of two chords of a circle is the ______ of the circle.
Answer: Centre
4. The circumcentre of any right-angled triangle is the ______ of the ______ of the triangle.
Answer: Midpoint of the hypotenuse
5. The circumcentre and incentre of an equilateral triangle are the ______ point.
Answer: Same
6. With respect to the circumcircle of a triangle, the sides of the triangle are ______ of the circle.
Answer: Chords
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Key Concepts from Chapter 11

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1. Circumcentre and Circumradius

The point where the perpendicular bisectors of the sides of a triangle meet is called the circumcentre. The distance from the circumcentre to any vertex of the triangle is the circumradius.

Circumradius = Distance from circumcentre to any vertex

2. Circumcentre of a Right Triangle

For a right-angled triangle, the circumcentre lies at the midpoint of its hypotenuse. This is one of the important results to remember while solving construction and geometry problems.

3. Equilateral Triangle

In an equilateral triangle, the circumcentre and the incentre coincide. Therefore, both centres are located at the same point.

4. Sides of a Triangle and Its Circumcircle

When a triangle is inscribed in its circumcircle, each side of the triangle forms a chord of the circle.

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Long Answer and Construction Questions

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Question 1: Construct the Circumcircle of a Triangle

Construct a triangle whose three sides are 5 cm, 7 cm and 8 cm. Then construct the circumcircle of the triangle.

Construction Procedure

Step 1: Draw triangle ABC such that BC = 8 cm, CA = 5 cm and AB = 7 cm.
Step 2: Construct the perpendicular bisector of side AB.
Step 3: Construct the perpendicular bisector of another side, such as BC.
Step 4: Let the two perpendicular bisectors meet at O. Point O is the circumcentre of triangle ABC.
Step 5: With O as centre and OA as radius, draw a circle. The circle will pass through A, B and C.
Result: The circle passing through A, B and C is the required circumcircle of triangle ABC.
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Question 2: Construct the Circumcircle of an Isosceles Triangle

Construct an isosceles triangle in which one angle is 120° and each of the equal sides is 4 cm. Then construct the circumcircle of the triangle.

Construction Procedure

Step 1: Construct triangle ABC such that BC = BA = 4 cm.
Step 2: Construct the required angle ∠ABC = 120°.
Step 3: Complete the isosceles triangle according to the given measurements.
Step 4: Draw the perpendicular bisectors of two sides of the triangle.
Step 5: Let their point of intersection be O. This point is the circumcentre.
Step 6: With O as centre and the distance from O to any vertex as radius, draw the circumcircle.
Result: The constructed circle passing through the three vertices is the required circumcircle of the isosceles triangle.
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Chapter 11 Quick Revision Chart

Topic Important Point
Incentre Centre of the incircle.
Inradius Radius of the incircle.
Circumcentre Centre of the circumcircle.
Circumradius Radius of the circumcircle.
Two Chords The perpendicular bisectors of two chords meet at the centre.
Right Triangle Circumcentre is the midpoint of the hypotenuse.
Equilateral Triangle Circumcentre and incentre are the same point.
Triangle in Circumcircle The three sides act as chords of the circle.

How to Prepare Chapter 11

  • Learn the definitions of incentre, inradius, circumcentre and circumradius.
  • Remember the special position of the circumcentre in a right-angled triangle.
  • Remember that the circumcentre and incentre coincide in an equilateral triangle.
  • Practise construction using perpendicular bisectors carefully.
  • For construction questions, maintain the given measurements accurately.
  • Revise the fill-in-the-blank questions because they test important terminology directly.

More Madhyamik Mathematics Resources

For broader WBBSE Class 10 preparation, explore the Madhyamik Suggestion 2027 – All Subjects resource on Cademy. It includes Mathematics along with the other Madhyamik subjects.

Revision Note: This chapter is especially useful for strengthening construction-based geometry. While practising, focus not only on the final figure but also on the correct sequence of construction steps.

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