Class 9 Maths Ganita Manjari Chapter 5 Exercise 5.2 Solutions

Class 9 Maths Ganita Manjari Chapter 5 Exercise 5.2 Solutions explain why a chord and the centre of a circle form an isosceles triangle.
These solutions also show how equal chord bases make two centre-chord triangles congruent by SSS.

Chapter 5, I’m Up and Down and Round and Round Class 9, develops circle geometry through centre, radius, chord, equal chords and triangle congruence. Class 9 Maths Ganita Manjari Chapter 5 Exercise 5.2 Solutions focus on two proof-based questions. Both questions use one simple fact: all radii of the same circle are equal. These Class 9 Maths Chapter 5 Exercise 5.2 Solutions help students write exact geometry proofs for the triangle formed by chord and centre, equal base lengths and SSS congruence circle chord questions. Exercise Set 5.2 appears after the theorems on equal chords and angles at the centre.

Key Takeaways

• Radius: Every radius of the same circle has equal length.
• Chord Triangle: A chord and the centre of a circle form an isosceles triangle.
• Equal Chords: Equal chords give equal bases in two centre-chord triangles.
• SSS Congruence: Two centre-chord triangles are congruent when all three corresponding sides are equal.

Class 9 Maths Ganita Manjari Chapter 5 Exercise 5.2 Solutions Structure 2026

Class 9 Maths Ganita Manjari Chapter 5 Exercise 5.2 Solutions for Chords and Centre

Exercise 5.2 contains two questions based on the triangle made by joining the centre of a circle to the endpoints of a chord. The exercise checks whether students can use equal radii and congruence rules in a short proof.

Q1. Show that the triangle formed by a chord and the centre of the circle is isosceles.

The triangle formed by a chord and the centre of the circle is isosceles because the two sides from the centre are radii of the same circle.

Given:

Let AB be a chord of a circle with centre O.

Join OA and OB.

To prove:

ΔOAB is an isosceles triangle.

Proof:

1. A lies on the circle.
2. B lies on the circle.
3. O is the centre of the circle.
4. OA is a radius of the circle.
5. OB is also a radius of the circle.

Equation:

OA = OB

Reason:

Radii of the same circle are equal.

Conclusion:

ΔOAB has two equal sides.

Therefore:

ΔOAB is an isosceles triangle.

Answer:

The triangle formed by a chord and the centre of the circle is isosceles because both sides from the centre to the chord endpoints are equal radii.

Ganita Manjari Class 9 Chapter 5 Exercise 5.2: Chord and Centre of Circle Isosceles Proof

A chord becomes the base of the triangle when its endpoints are joined to the centre. The two remaining sides are radii, so the triangle has two equal sides.

Concept Used in Question 1

Let AB be a chord.

Let O be the centre.

Join OA and OB.

Equation:

OA = OB

Reason:

OA and OB are radii of the same circle.

Conclusion:

ΔOAB is isosceles.

This is the direct proof pattern for chord and centre of circle isosceles questions.

Class 9 Maths Chapter 5 Exercise 5.2 Solutions Using SSS Congruence

Two centre-chord triangles in the same circle already have two pairs of equal sides because they contain radii. If their chord bases are also equal, the triangles become congruent by SSS.

Q2. Show that if two such isosceles triangles, occurring in the previous question, have equal base length, they are congruent to each other.

The two isosceles triangles are congruent by SSS because their two radius sides are equal and their base chord lengths are equal.

Given:

Let AB and CD be two chords of the same circle with centre O.

Join OA, OB, OC and OD.

Suppose:

AB = CD

To prove:

ΔOAB ≅ ΔOCD

Proof:

1. OA and OC are radii of the same circle.
2. OB and OD are also radii of the same circle.

Equations:

OA = OC

OB = OD

1. The base lengths are equal.

Equation:

AB = CD

1. The three sides of ΔOAB are equal to the three corresponding sides of ΔOCD.

Equations:

OA = OC

OB = OD

AB = CD

1. Therefore, by SSS congruence:

ΔOAB ≅ ΔOCD

Answer:

The two isosceles triangles are congruent by SSS congruence because all three corresponding sides are equal.

Class 9 Maths Ganita Manjari Chapter 5 Solutions: Equal Chords and Centre Angles

The section before Exercise 5.2 proves that equal chords subtend equal angles at centre. It also proves the converse, which says chords subtending equal angles are equal.

Equal Chords Subtend Equal Angles at Centre

If AB and DE are equal chords of a circle with centre C, then the angles at the centre are equal.

Given:

AB = DE

Radii:

CA = CB

CD = CE

Since all are radii of the same circle:

CA = CD

CB = CE

Using SSS congruence:

ΔCAB ≅ ΔCDE

Therefore:

∠ACB = ∠DCE

Answer:

Equal chords subtend equal angles at the centre because the two triangles formed with the centre are congruent by SSS.

Chords Subtending Equal Angles Are Equal

If two chords subtend equal angles at the centre, the chords are equal.

Given:

∠ACB = ∠DCE

Radii:

AC = DC

BC = EC

Using SAS congruence:

ΔACB ≅ ΔDCE

Therefore:

AB = DE

Answer:

Chords subtending equal angles are equal because the two triangles formed with the centre are congruent by SAS.

I’m Up and Down and Round and Round Class 9: Concepts Used in Exercise 5.2

Exercise 5.2 uses the circle definition that every point on a circle is at the same distance from the centre. Because of this, the two segments drawn from the centre to the endpoints of a chord are equal radii.

Triangle Formed by Chord and Centre

If AB is a chord and O is the centre, joining OA and OB forms ΔOAB.

Equation:

OA = OB

Reason:

OA and OB are radii of the same circle.

Conclusion:

ΔOAB is an isosceles triangle.

Equal Chords and Congruent Centre-Chord Triangles

If AB and CD are equal chords of the same circle with centre O, then the triangles formed with the centre are congruent.

Equations:

OA = OC

OB = OD

AB = CD

Therefore:

ΔOAB ≅ ΔOCD

Congruence rule:

SSS congruence

This is the exact proof method used in Ganita Manjari Class 9 Chapter 5 Exercise 5.2.

Class 9 Maths Circles Solutions: Proof Pattern for Exercise 5.2

Exercise 5.2 answers become easy when students first mark the equal radii. After that, the required result follows from the definition of an isosceles triangle or from the SSS congruence rule.

Proof Pattern for Q1

Use this order:

1. Name the chord.
2. Name the centre.
3. Join the centre to both endpoints of the chord.
4. State that both joining segments are radii.
5. Write the equality.
6. Conclude that the triangle is isosceles.

Copy-friendly proof:

OA = OB

Therefore, ΔOAB is isosceles.

Proof Pattern for Q2

Use this order:

1. Name both chords.
2. Join the centre to the endpoints of both chords.
3. Mark equal radii.
4. Use the given equal chord bases.
5. Apply SSS congruence.

Copy-friendly proof:

OA = OC

OB = OD

AB = CD

Therefore, ΔOAB ≅ ΔOCD by SSS congruence.

Quick Concept Table for Exercise 5.2

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 5