Question

The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. The area, T, of the triangle and the area, $R$, of the rectangle, both in sq cm, satisfy the relationship $R = T^2$ . If the sides of the rectangle are in the ratio $1: 3$, then the length, in cm, of the longer side of the rectangle, is

• A. 24
• B. 27
• C. 21
• D. 18

Answer 1

Answer: (B)

27

Answer 2

The correct answer is (B): $27$
Let the breadth of the rectangle be $b$.
Length of the rectangle = $3b$
Let $a$ be the side of the equilateral triangle.
Given,
$2(4b)+3a=90$
$⇒ 8\bigg(\frac{a^2}{4}\bigg)+3a-90 = 0$
$⇒ 2a^2+3a-90 = 0$
$⇒ (a-6)(2a+15) = 0$
$⇒ a = 6$
$∴ b = 9$
$⇒ 3b = 27$