Q-1 Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

Solution:

Given

Two circles with centres P and R, which intersect each other at Q and S

Construction

Join PQ, PS, SR, QR.

In and

(radius of the same circle)

(radius of the same circle)

(common line)

Therefore by Side-Side-Side criterion of congruence,

Therefore, (corresponding parts of congruent triangles)

Q-2 Two chords PQ and RS of lengths 5cm and 11cm respectively of a circle are parallel to each and are on opposite sides of its centre. If the distance between PQ and RS is 6cm, find the radius of the circle

Solution:

Given,

O is the centre of the circle

And its radius be r cm

and

Construction,

Join OP and OR

Since both P and R are on the circle and O is the center,

and and , Therefore, B, O, A are collinear.

Let , since . Therefore,

Since, the perpendicular from the centre to a chord of the circle bisects the chord, therefore,