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, find x.

Answer 1

Given:

Quadrilateral ABCD circumscribes a circle; sides AB, BC, CD, AD touch the circle at P, Q, R, S respectively.

AB = x cm, BC = 7 cm, CR = 3 cm, AS = 5 cm.

Use the property: tangents drawn from the same external point to a circle are equal in length.

Step-wise calculation:

1. From A: AP = AS.

Since AS = 5 cm, AP = 5 cm.

2. From C: CQ = CR.

Since CR = 3 cm, CQ = 3 cm.

3. BC = BQ + CQ = 7 cm.

So BQ = 7 – CQ

= 7 – 3

= 4 cm

4. From B: BP = BQ.

So BP = 4 cm.

5. AB = AP + PB

= 5 + 4

= 9 cm

Hence x = 9.