NCERT Solutions Class 9 Maths
Chapter – 10 (Circles)
The NCERT Solutions in English Language for Class 9 Mathematics Chapter – 10 Circles Exercise 10.2 has been provided here to help the students in solving the questions from this exercise.
Chapter 10: Circles
- NCERT Solution Class 9 Maths Ex – 10.1
- NCERT Solution Class 9 Maths Ex – 10.3
- NCERT Solution Class 9 Maths Ex – 10.4
- NCERT Solution Class 9 Maths Ex – 10.5
- NCERT Solution Class 9 Maths Ex – 10.6
Exercise – 10.2
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.
Answer – Let QR and YZ be the equal chords of 2 congruent circles.
Let QR and YZ be the equal chords of 2 congruent circles.
Then, QR = YZ
We need to prove that they subtend equal angles at the center. That is, ∠QPR = ∠YXZ
We know that the radii of both circles are equal. So, we get: PR = PQ = XZ = XY
Consider the 2 triangles, ∆PQR and ∆XYZ.
PQ = XY (Radii are equal)
PR = XZ (Radii are equal)
QR = YZ (Chords are equal)
By SSS congruency, ∆PQR is congruent to ∆XYZ.
So, by CPCT (Corresponding parts of congruent triangles), we get ∠QPR =∠YXZ.
Hence, proved that equal chords of congruent circles subtend equal angles at their centers.
2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
Answer – Using equal angles at the centers and the fact that circles are congruent, we prove the statement using Side-Angle-Side (SAS criteria) and corresponding parts of congruent triangles (CPCT).
Draw chords QR and YZ in two congruent circles as shown above. Join the radii PR, PQ, and XY, XZ respectively.
Given that chords subtend equal angles at the center. So, ∠QPR = ∠YXZ.
We need to prove that chords are equal, that is, QR = YZ
Since the circles are congruent, their radii will be equal.
PR = PQ = XZ = XY
Consider the two triangles ∆PQR and ∆XYZ.
PQ = XY (Radii are equal)
∠QPR = ∠YXZ (Chords subtend equal angles at center)
PR = XZ (Radii are equal)
By SAS criteria, ∆PQR is congruent to ∆XYZ.
So, QR = YZ (Corresponding parts of congruent triangles)
Hence proved if chords of congruent circles subtend equal angles at their center then the chords are equal.
