In the figure (i) given below, AB is a diameter of a circle with center O. AC and BD are perpendiculars on a line PQ. BD meets the circle at E. Prove that AC = ED.

Join AE.

∠AEB = 90° (∵ angle in semicircle is 90°.)

∠AED = 90° (∵ ∠AEB and ∠AED form a linear pair.)

Hence, we can say that,

DE is also perpendicular to AE, since DE is also perpendicular to PQ hence,

AE || PQ.

Since, CA and DE both are perpendicular to PQ hence,

CA || DE.

Hence, proved that ACDE is a rectangle.

In rectangle opposite sides are equal so,

AC = DE.

Hence, proved that AC = DE.