Question

In Fig. a quadrilateral ABCD is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch the circle at P, Q, R and S respectively. If AB = x cm, BC = 7 cm, CR = 3 cm and AS =5 cm, then x=

A. 10 

B. 9 

C. 8 

D. 7

Answer 1

Answer is B. 9

Given: 

AB = x cm 

BC = 7 cm 

CR = 3 cm 

AS =5 cm 

Property: If two tangents are drawn to a circle from one external point, then their tangent segments (lines joining the external point and the points of tangency on circle) are equal. 

By the above property, 

AP = AS (tangent from A) 

BP = BQ (tangent from B) 

CR = CQ (tangent from C) 

DR = DS (tangent from D) 

Clearly, 

QB = CB – CQ 

⇒ QB = CB – CR [∵ CQ = CR] 

⇒ QB = 7 cm – 3 cm 

⇒ QB = 4 cm 

Now, 

AB = AP + PB 

⇒ AB = AS + QB

⇒ AB = 5 cm + 4 cm 

⇒ AB = 9 cm 

⇒ AB = x = 9 cm 

Hence, x = 9 cm