Q. 224.5( 76 Votes )

# Prove that the re

Answer : Given – ABCD is a rectangle circumscribed in circle with centre O.

To prove – ABCD is a square.

Property – Lengths of the two tangents drawn from an external point to a circle are equal.

Answer –

We know that, opposite sides of a rectangle are equal.

AB = CD & AD = BC ………(1)

As lengths of the two tangents drawn from an external point to a circle are equal.

AP = AS ………(2)

BP = BQ ………(3)

CR = CQ ………(4)

DR = DS ………(5)

Adding (1), (2), (3) & (4),

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

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

AB + CD = AD + BC ………from figure

AB + AB = BC + BC ………from (1)

2AB = 2BC

AB = BC

Therefore, adjacent sides of ABCD are equal.

Rectangle with equal adjacent sides is a square.

Hence, ABCD is a square.

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