Question

In the given figure, 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 [CBSE 2014]

Answer 1

It is given that 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.

Also, BC = 7 cm, CR = 3 cm and AS = 5 cm

Now, AP and AS are tangents drawn to the same circle from an external point A.

∴ AP = AS = 5 cm .....(1) (Lengths of tangents drawn from an external point to a circle are equal)

CQ and CR are tangents drawn to the same circle from an external point C.

∴ CQ = CR = 3 cm (Lengths of tangents drawn from an external point to a circle are equal)

So, BQ = BC − CQ = 7 cm − 3 cm = 4 cm

BP and BQ are tangents drawn to the same circle from an external point B.

BP = BQ = 4 cm .....(2) (Lengths of tangents drawn from an external point to a circle are equal)

Now,

AB = AP + PB = 5 cm + 4 cm = 9 cm [From (1) and (2)]

∴ x = 9 (AB = x cm)

Hence, the correct answer is option B.