Question:medium

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

Updated On: Jan 15, 2026
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The Correct Option is B

Solution and Explanation

Let the breadth of the rectangle be \( b \). 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 by the equation: \( 2 \times \text{Perimeter of Rectangle} + 3 \times \text{Side of Triangle} = 90 \). Substituting the given dimensions: \( 2(4b) + 3a = 90 \), which simplifies to \( 8b + 3a = 90 \) (Equation 1).

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

Using the provided area relationship: \( b = \frac{a^2}{4} \). Substitute this into Equation 1: \( 8 \left( \frac{a^2}{4} \right) + 3a = 90 \). This simplifies to \( 2a^2 + 3a = 90 \), or \( 2a^2 + 3a - 90 = 0 \).

Step 3: Solve the Quadratic Equation

Solve the quadratic equation \( 2a^2 + 3a - 90 = 0 \) by factorization: \( (2a + 15)(a - 6) = 0 \). Since the side length \( a \) must be positive, \( a = 6 \).

Step 4: Compute Breadth and Length

Calculate the breadth \( b \) using \( b = \frac{a^2}{4} = \frac{6^2}{4} = \frac{36}{4} = 9 \). The length is \( 3b = 3 \times 9 = 27 \).

✅ Final Answer: Length = 27 units

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