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CAT 2022 Slot 1 — QA Question 16

Basics of CirclesEasy

All the vertices of a rectangle lie on a circle of radius R. If the perimeter of the rectangle is P, then the area of the rectangle is

Answer & solution

  • A

    P22 - 2PR

  • P28 - 2R2

  • C

    P28 - R22

  • D

    P216 - R2

Solution

Easy

A rectangle inscribed in a circle has the diameter as its diagonal, so a2+b2=(2R)2a^2+b^2=(2R)^2. The perimeter gives a+ba+b. Square a+ba+b and the area abab pops out via (a+b)2=a2+b2+2ab(a+b)^2=a^2+b^2+2ab.

a b 2R
1

Use the diagonal. The rectangle's diagonal is a diameter, so with sides a,ba,b:

a2+b2=(2R)2=4R2a^2+b^2=(2R)^2=4R^2
2

Use the perimeter.

P=2(a+b) ⇒ a+b=P2P=2(a+b)\ \Rightarrow\ a+b=\frac{P}{2}
3

Square a+ba+b to bring in abab:

(a+b)2=a2+b2+2abP24=4R2+2ab\begin{aligned} (a+b)^2&=a^2+b^2+2ab\\ \frac{P^2}{4}&=4R^2+2ab \end{aligned}
4

Solve for the area abab.

2ab=P24−4R2ab=P28−2R2\begin{aligned} 2ab&=\frac{P^2}{4}-4R^2\\ ab&=\frac{P^2}{8}-2R^2 \end{aligned}
Area=ab=P28−2R2 — option (b)\text{Area}=ab=\frac{P^2}{8}-2R^2\ \text{— option (b)}
CAT 2022 Slot 1 QA Q16: All the vertices of a rectangle lie on a circle of radius R. If the perimeter of the rectangle is P, then the — Solution | TheCATExam