State True or False.

The rectangle circumscribing a circle is a square.

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True

False

answer is A.

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Detailed Solution

Given a rectangle circumscribing a circle.
Consider ABCD is a rectangle circumscribed in a circle with center O.
Question Image

We know that the opposite sides of a rectangle are equal.
Here, AB = CD & AD = BC ………(1)
Also, we know that lengths of the two tangents drawn from an external point to a circle are equal.
Therefore, from the figure we get,
AP = AS ………(2)
BP = BQ ………(3)
CR = CQ ………(4)
DR = DS ………(5)
Add eq 2,3,4 and 5.
We get,
AP + BP + CR + DR = AS + BQ + CQ + DS
(AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ) …….(6)
We can observe from the figure,
AP + BP = AB
CR + DR = CD
AS + DS = AD
BQ + CQ = BC
Substitute the above values in eq (6).
AB + CD = AD + BC
From eq (1), AB = CD & AD = BC.
AB + AB = BC + BC
2AB = 2BC
AB = BC
Here, from above equation we can say that adjacent sides of ABCD are equal.
We know that square is a rectangle with equal adjacent sides.
Hence, ABCD is a square.
So, the given statement is true.
The correct option is (1).

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