Question

A straight line is drawn cutting two equal circles and passing through the midpoint M of the line joining their centres O and O’. Prove that chords AB and CD, which are intercepted by the two circles are equal.

Answer 1

Given –
A straight line AD intersects two circles of equal radii at A, B, C and D.
Line joining the centres OO’ intersect AD at M
M is the midpoint of OO’
To prove –: AB = CD.
Construction – From the centre O, draw OP ⊥ AB and from O’ draw O’Q ⊥ CD.
Proof –:
In $\Delta$ OMP and $\Delta$ O’MQ,
∠ OMP = ∠ O’MQ [vertically opposite angles]
∠ OPM = ∠ O’QM [each = 90${}^o$]
OM = O’M [given]
By AAS criterion of congruence,
$\Delta OMP\cong \Delta O^{\prime }MP$
OP = O’Q [c.p.c.t]
We know that, two chords of a circle or equal circles which are equidistant from the centre are equal.
Hence, AB = CD.