In â ABCD, m∠D = 90. A circle with centre 0 and radius r touches its sides and in P, Q, R and S respectively. If BC = 40, CD = 30 and BP = 25, then find the radius of the circle.

Given that in â ABCD, ∠D = 90°. BC = 40, CD = 30 and BP = 25

We know that tangents drawn to a circle are perpendicular to the radius of the circle.

⇒ ∠ORD = ∠OSD = 90°

Given ∠D = 90° and OR = OS = radius.

∴ ORDS is a square.

We know that the tangents drawn to a circle from a point in the exterior of the circle are congruent.

∴ BP = BQ, CQ = CR and DR = DS.

Consider BP = BQ,

⇒ BQ = 25 [BP = 25]

⇒ BC – CQ = 40

⇒ CQ = 40 – 25 = 15 [BC = 40]

Consider CQ = CR,

⇒ CR = 15

⇒ CD – DR = 15

⇒ DR = 30 – 15 = 15 [CD = 30]

But ORDS is a square.

∴ OR = DR = 15

∴ Radius of circle OR is 15.