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

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Step 1: Rectangle ABCD circumscribing a circle with centre O, tangent points P, Q, R, S on sides AB, BC, CD, DA
Step 2: $AP = AS$, $BP = BQ$, $CR = CQ$, $DR = DS$.
Adding: $AB + CD = AD + CB$.
Step 3: Since ABCD is a rectangle: $AB = CD$ and $BC = AD$.
$\Rightarrow 2AB = 2AD \Rightarrow AB = AD$. Hence ABCD is a square.
Step 4: Side of square $=$ diameter of circle $= 20$ cm.
Perimeter $= 4 	imes 20 = 80$ cm.

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