Class 9 NCERT Solutions - Chapter 10 Circles - Exercise 10.2

Chapter 10 of the Class 9 NCERT Mathematics textbook delves into the geometric properties of circles. Exercise 10.2 focuses on the problems related to angles subtended by chords and arcs in a circle.

Understanding these concepts is crucial for the solving problems involving the relationships between the different parts of a circle and their properties.

What is a Circle?

A circle is a simple closed curve in a plane that is equidistant from the fixed point known as the center. The distance from the center to any point on the circle is called the radius and line segment that passes through the center and connects two points on the circle is called the diameter.

A circle can be divided into the several segments and sectors and relationships between these parts are fundamental in geometry. Key terms related to circles include the chord, arc and sector.

Question 1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres. 

Solution:

👁 Image

Given: 

Two Congruent Circles C1 and C2

AB is the chord of C1

and PQ is the chord of C2

AB = PQ

To Prove: Angle subtended by the Chords AB and PQ are equal i.e. ∠AOB = ∠PXQ

Proof:

In △AOB & △PXQ

AO = PX                    (Radius of congruent circles are equal)

BO = QX                      (Radius of congruent circles are equal)

AB = PQ                    (Given)

△AOB ⩭ △PXQ       (SSS congruence rule)

Therefore, ∠AOB = ∠PXQ         (CPCT)

Question 2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Solution: 

👁 Image

Given:

Two Congruent circles C1 and C2

AB is the chord of C1 and PQ is chord of C2

& ∠AOB = ∠PXQ  

To prove : 

In △AOB and  △PXQ ,

AO = PX                         (Radius of congruent circles are equal)

∠AOB = ∠PXQ              (Given)

BO = QX                          (Radius of congruent circles are equal)

△AOB ⩭ △PXQ          (SAS congruence rule)

Therefore, AB = PQ       (CPCT)

Conclusion

Exercise 10.2 in Chapter 10 helps students apply their understanding of circles by the solving problems related to angles subtended by the chords and arcs. Mastery of these concepts is essential for the grasping more complex geometric principles and solving related problems efficiently. By working through these exercises students enhance their analytical skills and gain a deeper insight into the properties of circles.