Question

In the figure, a circle is inscribed in a quadrilateral ABCD in which ∠B = 90°. If AD = 23 cm, AB = 29 cm and DS = 5 cm, find the radius r of the circle.

Answer 1

Given AD = 23 cm

AB = 29 cm

∠B = 90°

DS = 5cm

From fig in quadrilateral POQB

∠OPB = ∠OQB = 90° = ∠B = ∠POQ

and PO = OQ. ∴ POQB is a square PB = BQ = r

We know that

Tangents drawn from external point to circle are equal in length.

We know that

Tangents drawn from external point to circle are equal in length.

From A, AR = AQ …. (i)

From B, PB = QB …. (ii)

From C, PC = CS …. (iii)

From D, DR = DS …. (iv)

(i) + (ii) + (iv) ⇒ AR + DB + DR = AQ + QB + DS

⇒ (AR + DR) + r = (AQ + QB) + DS

AD + r = AB + DS

⇒ 23 + r = 29 + 5

⇒ r = 34 – 23 = 11 cm

∴ radius = 11 cm