Question

In Fig. 3, AP and BP are tangents to a circle with centre O, such that AP = 5 cm and ∠APB = 60°. Find the length of chord AB.

Answer 1

PA and PB are tangents drawn to the given circle from an external point P.

It is known that the lengths of the tangents drawn from an external point to a circle are equal.

∴ PA = PB

In ∆PAB, sides PA and PB are of the same length.

Hence, ∆PAB is isosceles, with PA = PB and ∠PAB = ∠PBA = x (say).

It is given that

∠APB = 60°

We know that the sum of the angles of a triangle is 180°

In ∆PAB,

∠PAB + ∠PBA + ∠APB = 180°

∴ x + x + 60° = 180°

⇒ 2x = 120°

⇒ x = 60°

Thus,

∠PAB = ∠PBA = ∠APB = 60°

Since all angles of ∆PAB are of the same measure, ∆PAB is equilateral, with AP = BP = AB.
It is given that

AP = 5 cm

∴ AB = AP = 5 cm

Thus, the length of the chord AB is 5 cm.