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

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.