Question 22 Prove that the rectangle circumscribing a circle is a square.
Given:
A circle with centre O.
A rectangle ABCD touching the
circle at points P, Q, R and S
To prove: ABCD is a square
Proof:
A rectangle is a square with all sides equal,
So, we have to prove all sides equal
We know that
lengths of tangents drawn from external point are equal
Hence,
AP = AS
BP = BQ
CR = CQ
DR = DS
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
…(1)
…(2)
…(3)
…(4)
Since ABCD is a rectangle,
Opposite sides are equal
∴ CD = AB & BC = AD
AB + AB = AD + AD
2AB = 2AD
AB = AD
So,
AB = AD
& AB = CD & AD = BC
So, AB = CD = AD = CD
So, ABCD is a rectangle with all sides equal
∴ ABCD is a square
Hence proved

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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