Exercise E10.2 Circles NCERT Solutions Class 9
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 centers.
Solution
Video Solution
What is known?
Two circles are congruent if they have same radii.
What is unknown?
Equal chords of congruent circles subtend equal angles at their centers.
Reasoning:
Using chords are equal and the fact that circles are congruent, we prove the statement using Side-Side-Side (SSS criteria) and Corresponding parts of congruent triangles (CPCT).
Steps:
Let \(\begin{align} {QR} \end{align}\) and \(\begin{align} {YZ} \end{align}\) be the equal chords of \(2\) congruent circles.
\(\begin{align} {QR = YZ} \end{align}\)
We need to prove that they subtend equal angles at centre. i.e. \(\begin{align}\angle{QPR} =\angle {YXZ}\end{align}\)
We know that the radii of both the circles are equal. So we get:
\(\begin{align} {PR = PQ = XZ = XY} \end{align}\)
Consider the \(2\) triangles \(\begin{align} \Delta { PQR} \end{align}\) and \(\begin{align} \Delta { XYZ} \end{align}\).
\(\begin{align} &{PQ = XY} \qquad \text{(Radii are equal)}\\&{PR = XZ} \qquad \text{(Radii are equal)}\\&{QR = YZ} \qquad \text{(Chords are equal)}\end{align}\)
By SSS criteria, \(\begin{align} \Delta\,{PQR}\end{align}\) is congruent to \(\begin{align} \Delta\;{XYZ}.\end{align}\)
So by CPCT (Corresponding parts of congruent triangles) we get \(\begin{align} \angle{QPR}=\angle{YXZ}.\end{align}\)
Hence proved that equal chords of congruent circles subtend equal angles at their centres.
Question 2
Prove that if chords of congruent circles subtend equal angles at their centers, then the chords are equal.
Solution
Video Solution
What is known?
Two circles are congruent, and their chords subtend equal angles at their centers.
What is unknown?
To find whether the chords are equal.
Reasoning:
Using equal angles at centres and the fact that circles are congruent, we prove the statement using Side-Angle-Side (SAS criteria) and Corresponding parts of congruent triangles (CPCT).
Steps:
Draw chords \(\begin{align} {QR}\end{align}\) and \(\begin{align} {YZ}\end{align}\) in \(2\) congruent circles respectively. Join the radii \(\begin{align} {PR}, \; {PQ} \end{align}\) and \(\begin{align} {XY, XZ }\end{align}\) respectively.
Given that chords subtend equal angles at centre. So\(\begin{align}\angle {QPR}=\angle {YXZ}\,.\end{align}\)
We need to prove that chords are equal. i.e \(\begin{align}{QR = YZ}.\end{align}\)
Since the circles are congruent, their radii will be equal.
\[\begin{align}{PR = PQ = XZ = XY}\end{align}\]
Consider the \(2\) triangles \(\begin{align} \Delta\,{PQR}\end{align}\) and \(\begin{align} \Delta\,{XYZ}.\end{align}\)
\(\begin{align}{{PQ}} & ={{XY}} \qquad \;\; \begin{pmatrix} \text { Radii } \\ \text{are equal }\end{pmatrix} \\ {\angle {QPR}} &={\angle {YXZ}} \quad \begin{pmatrix} \! \text { Chords } \! \\ \! \text{subtend} \! \\ \! \text{equal angles} \! \\ \! \text{ at centre } \! \end{pmatrix} \\ {{PR}} &= {{XZ}} \qquad \;\; \begin{pmatrix} \text { Radii } \\ \text{are equal }\end{pmatrix} \end{align}\)
By SAS criteria \(\begin{align} \Delta { PQR}\end{align}\) is congruent to \(\begin{align} \Delta { XYZ.}\end{align}\)
So by CPCT (Corresponding parts of congruent triangles) \(\begin{align} {QR = YZ.}\end{align}\)
Hence proved if chords of congruent circles subtend equal angles at their centres then the chords are equal.