Question

Rectangle ABCD circumscribes the circle of radius 10 cm. Prove that ABCD is a square. Hence, find the perimeter of ABCD.

Answer 1

ABCD is a rectangle.

∴ AB = DС  ...(1)

AD = BC  ...(2)

AP = AS, PB = BQ

CR = CQ, DR = DS   ...(∵ Tangents to a circle)

On adding the above,

AP + PB + CR + DR = AS + BQ + CQ + DS

⇒ (AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)

⇒ AB + CD = AD + BС

From equations (1) and (2),

AB + AB = AD + AD

2AB = 2AD

AB = AD

AB = DC = AD = ВС

Hence, ABCD is a square.

Hence proved.

Diameter of circle = 2 × 10

= 20 cm

AB = BC = CD = AD = PR = SQ = 20 cm

Perimeter of square = 4a

= 4 × 20

= 80 cm