CAT 2020 Slot 2 QA Question 11: The sum of 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

Question

The sum of 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

Accepted Answer

Let the side of the equilateral triangle be $a$ cm. Let the rectangle sides be $x$ and $3x$ cm.

Using perimeters,
$3a + 8x = 90$

The areas are
$T = \dfrac{\sqrt{3}}{4} a^2$
$R = 3x^2$

Given $R = T^2$,
$3x^2 = \left(\dfrac{\sqrt{3}}{4} a^2\right)^2$
$3x^2 = \dfrac{3}{16} a^4$
$x = \dfrac{a^2}{4}$