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 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Hi, it looks like you're using AdBlock :(

Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription.

Please login to view more pages. It's free :)

Teachoo gives you a better experience when you're logged in. Please login :)

Solve all your doubts with Teachoo Black!

Teachoo answers all your questions if you are a Black user!