Circles ⇒⇒ Exercise 9.1

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 :

Given : Two circles having centres O and O’ and having equal radius.

Let AB and CD be the equal chords of 2 congruent circles.

AB = CD

We want to prove that the angles subtended by the chords at the centers are equal, i.e.,

$\angle $ AOB = $\angle $ CO’D

In Δ AOB and Δ CO'D

We know that the radius of both the circles are equal. So ,

OA = O’C

OB = O’D

( Equal radius of the congruent circles )

AB = CD

( Given equal chords )

Since all three corresponding sides of Δ AOB and Δ CO'D are equal, the two triangles are congruent by the SSS congruence criterion.

Δ AOB ≅ Δ CO'D

By the property of Corresponding Parts of Congruent Triangles (CPCT), the corresponding angles of the congruent triangles must be equal. Therefore,

∴ $\angle $ AOB = $\angle $ CO’D

Hence Proved.

Question 2

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

Solution :

Given that two circles are congruent with centers O and O' and having equal radius..

Let chords of congruent circles subtend equal angles at their centres.

$\angle $ AOB = $\angle $ CO’D

To Prove: AB and CD be the equal chords of 2 congruent circles.

AB = CD

In Δ AOB and Δ CO'D

We know that the radius of both the circles are equal. So ,

OA = O’C

( Equal radius of the congruent circles )

$\angle $ AOB = $\angle $ CO’D

( Chords subtend equal angles at centre )

OB = O’D

( Equal radius of the congruent circles )

Since two corresponding sides and the included angle of Δ AOB and Δ CO'D are equal, the two triangles are congruent by the SAS congruence criterion.

Δ AOB ≅ Δ CO'D

By the property of Corresponding Parts of Congruent Triangles (CPCT), the corresponding angles of the congruent triangles must be equal. Therefore,

∴ AB = CD

Hence Proved.