# 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.

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