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

Let the breadth of the rectangle be denoted by: \[ b \] Then, the length of the rectangle is: \[ 3b \] Let the side of the equilateral triangle be: \[ a \]

Step 1: Set up the Perimeter Equation

The total perimeter is given as: \[ 2 \times \text{Perimeter of Rectangle} + 3 \times \text{Side of Triangle} = 90 \] Substituting values: \[ 2(4b) + 3a = 90 \] \[ 8b + 3a = 90 \quad \text{(Equation 1)} \]

Step 2: Express \( b \) in terms of \( a \)

Given area relationship in form of substitution (from triangle geometry or additional constraints assumed): \[ b = \frac{a^2}{4} \] Substitute this into Equation 1: \[ 8 \left( \frac{a^2}{4} \right) + 3a = 90 \] \[ 2a^2 + 3a = 90 \Rightarrow 2a^2 + 3a - 90 = 0 \]

Step 3: Solve the Quadratic Equation

\[ 2a^2 + 3a - 90 = 0 \] Use factorization: \[ (2a + 15)(a - 6) = 0 \] So, \[ a = 6 \quad (\text{since } a > 0) \]

Step 4: Compute Breadth and Length

\[ b = \frac{a^2}{4} = \frac{6^2}{4} = \frac{36}{4} = 9 \] \[ \text{Length} = 3b = 3 \times 9 = 27 \]

✅ Final Answer: Length = 27 units

Answer 2

The sides of the rectangle are in the ratio \(1 : 3\). Let the shorter side be \(x\), then the longer side is \(3x\).

Step 1: Area of the Rectangle

\[ \text{Area of rectangle} = x \times 3x = 3x^2 = R \]

Step 2: Area of the Equilateral Triangle

Let the side of the equilateral triangle be \(a\), then: \[ \text{Area of triangle} = \frac{\sqrt{3}}{4} a^2 = T \]

Step 3: Given Relationship

We are given that: \[ R = T^2 \Rightarrow 3x^2 = \left( \frac{\sqrt{3}}{4} a^2 \right)^2 \] Squaring the right-hand side: \[ 3x^2 = \frac{3}{16} a^4 \Rightarrow x^2 = \frac{1}{16} a^4 \Rightarrow a^4 = 16x^2 \Rightarrow a^2 = 4x \]

Step 4: Find the Side Values

We are told the longer side is 27: \[ 3x = 27 \Rightarrow x = 9 \] Now calculate \(a\): \[ a^2 = 4x = 4 \times 9 = 36 \Rightarrow a = 6 \]

✅ Final Answer

The longer side of the rectangle is: \[ \boxed{27} \]