#### Question

A straight line is drawn cutting two equal circles and passing through the mid-point M of the line joining their centres O and O’

Prove that the chords AB and CD, which are intercepted by the two circles are equal.

#### Solution

Given: A straight line Ad intersects two circles of equal radii at A, B, C and D.

The line joining the centres O O' intersect AD at M

And M is the midpoint of OO'.

To prove: AB = CD

Construction: From O, draw OP⊥AB and from O’, draw O 'Q⊥CD.

Proof:

In ΔOMP and Δ O' MQ,

∠OMP = ∠O'MQ (vertically opposite angles)

∠OPM = ∠O'QM (each = 90°)

OM = O'M (Given)

By Angle – Angle – Side criterion of congruence,

∴ ΔOMP ≅ O'MQ, (by AAS)

The corresponding parts of the congruent triangle are congruent

∴ OP = O'Q (c.p.ct)

We know that two chords of a circle or equal circles which are equidistant from the centre are equal.

∴ AB = CD