Question

A straight line is drawn cutting two equal circles and passing through the mid-point M of the line joining their centers O and O'. Prove that the 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.
The line joining the centers OO' 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 ΔO'MQ,
∠OMP = ∠O'MQ            ...( Vertically Opposite angles )
∠OPM = ∠O'QM            ...( each = 90° ) 
OM = O'M                      ...( Given )

By Angle-Angle-Side criterion of congruence,
∴ ΔOMP ≅ ΔO'MQ,       ...( by AAS )

The corresponding parts of the congruent triangles are congruent.
∴ OP = O'Q                     ...( c.p.c.t. )

We know that two chords of a circle or equal circles which are equidistant from the center are equal.
∴ AB = CD.