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CAT 2020 Slot 2 — QA Question 12

Basics of QuadrilateralsEasy

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 = T². 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

Answer & solution

  • 27

  • B

    24

  • C

    18

  • D

    21

Solution

Easy

Name the triangle side and the rectangle's shorter side. Write the perimeter equation, then turn the area condition R=T2R=T^2 into a relation between the two unknowns, substitute, and solve the resulting quadratic.

1

Set up the variables. Let the equilateral triangle have side aa, so its perimeter is 3a3a. Let the rectangle's sides be xx and 3x3x (ratio 1:31:3), so its perimeter is 2(x+3x)=8x2(x+3x)=8x.

3a+8x=90(sum of perimeters)\begin{aligned} &3a + 8x = 90 \quad\text{(sum of perimeters)} \end{aligned}
2

Use the area condition R=T2R=T^2. The triangle's area is T=34a2T=\tfrac{\sqrt3}{4}a^2 and the rectangle's area is R=x⋅3x=3x2R=x\cdot 3x=3x^2.

3x2=(34a2)2(R=T2)⇒ 3x2=316a4⇒ 16x2=a4(×163)⇒ 4x=a2(positive square root)\begin{aligned} &3x^2 = \left(\tfrac{\sqrt3}{4}a^2\right)^2 \quad\text{(}R=T^2\text{)}\\ &\Rightarrow\ 3x^2 = \tfrac{3}{16}a^4\\ &\Rightarrow\ 16x^2 = a^4 \quad\text{(}\times \tfrac{16}{3}\text{)}\\ &\Rightarrow\ 4x = a^2 \quad\text{(positive square root)} \end{aligned}
3

Substitute and solve. From step 2, x=a24x=\tfrac{a^2}{4}, so 8x=2a28x=2a^2. Put this into the perimeter equation of step 1.

3a+2a2=90⇒ 2a2+3a−90=0⇒ (2a+15)(a−6)=0⇒ a=6(reject a=−152)\begin{aligned} &3a + 2a^2 = 90\\ &\Rightarrow\ 2a^2 + 3a - 90 = 0\\ &\Rightarrow\ (2a+15)(a-6)=0\\ &\Rightarrow\ a = 6 \quad\text{(reject }a=-\tfrac{15}{2}\text{)} \end{aligned}
4

Find the longer side. With a=6a=6, get xx then the longer side 3x3x.

x=a24=364=9⇒ 3x=27\begin{aligned} &x = \tfrac{a^2}{4} = \tfrac{36}{4} = 9\\ &\Rightarrow\ 3x = 27 \end{aligned}
Longer side=27 cm\text{Longer side} = 27\ \text{cm}
CAT 2020 Slot 2 QA Q12: The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. The area, T, of the triangle an — Solution | TheCATExam