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

The Correct Option is B

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,
\(2(4b)+3a=90\)
\(⇒ 8\bigg(\frac{a^2}{4}\bigg)+3a-90 = 0\)
\(⇒ 2a^2+3a-90 = 0\)
\(⇒ (a-6)(2a+15) = 0\)
\(⇒ a = 6\)
\(∴ b = 9\)
\(⇒ 3b = 27\)

Answer 2

Given, Sides of the rectangle are in the ratio 1 : 3
Take as x and 3x
Area of Rectangle = 3x² = R
Take an equilateral triangle with side a
\(Area = \frac{√3}{4}× a^2 =T\)
Given R = T²
\(3x^2= \frac{√3}{4}× a^2\)
Solving we get a² = 4x
Longer side 3x = 27
x = 9. 
Sides of the rectangle are 9 and 27.
If x = 9, Then a = 6
Checking if a² = 4x
6² = 4(9)
Hence longer side of rectangle is 27