In the following figure, the line ABCD is perpendicular to PQ; where P and Q are the centers of the circles. Show that :

(i) AB = CD

(ii) AC = BD

We know that,

Perpendicular from center to chord, bisects the chord.

Since, ABCD is perpendicular to PQ.

∴ OA = OD …………(1)

Also,

⇒ OB = OC …………..(2)

(i) Subtracting equation (2) from (1), we get :

⇒ OA - OB = OD - OC

⇒ AB = CD.

Hence, proved that AB = CD.

(ii) We know that,

⇒ AB = CD

Adding BC to both sides of above equation, we get :

⇒ AB + BC = CD + BC

⇒ AC = BD.

Hence, proved that AC = BD.