PROVE THAT THE RECTANGLE CIRCUMSCRIBING A CIRCLE IS SQUARE.
Answers
Step-by-step explanation:
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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Answer:
Hence,ABCD IS A SQUARE
