Question

In the given figure, a circle with center O, is Inscribed in quadrilateral ABCD such that it touches the sides AB, BC, CD and AD at point P, Q, R and S respectively. If radius of circle ¡s 10 cm, BC = 38 cm, PB = 27 cm and AD ⊥ CD, then find the length of CD. 

Answer 1

DR and DS are the tangents from point D and OS and OR are the radius of circle.

∴ AD ⊥ OS and DR ⊥ OR 

AD ⊥ CD (given)

In quadrilateral DROS

∠D + ∠R + ∠O + ∠S = 360°

⇒ 90° + 90° + ∠O + 90° = 360°

⇒ ∠O = 360° – 270° = 90°

Similarly in quadrilateral DROS

∠D = ∠R = ∠O = ∠S = 90°

and OS = OR [radii of a circle

So, DROS is a square

So, SD = DR = 10 cm (Tangents on circle from point D.)

∵ Tangents BP and BQ on circle from point B.

∴ BP = BQ = 27 cm

CQ = BC – BQ

⇒ CQ = 38 – 27 = 11 cm

∵ CR and EQ are the tangents of circle from point C.

∴ CR = CQ

⇒ CR = 11 cm

Now CD = CR + DR

⇒ CD = 11 + 10

⇒ CD = 21 cm