The lengths of the parallel sides of a trapezium are 11 cm and 25 cm and the…

Concept: The area of a trapezium with parallel sides a and b and perpendicular height h between them is Area = ½ × (a + b) × h. When two plane figures are stated to have equal areas, their area expressions can be equated directly; and when a rectangle's two sides are in the ratio m : n, the sides can be represented as mx and nx (for some positive x), so the rectangle's area becomes mx × nx = mn × x2.

Application:

1. Area of the given trapezium = ½ × (11 + 25) × 12 = ½ × 36 × 12 = 216 cm2.

2. The rectangle has this same area, and its sides are in the ratio 3 : 2, so let its sides be 3x cm and 2x cm.

3. Equating areas: 3x × 2x = 6x2 = 216, so x2 = 36 and x = 6.

4. So the rectangle's sides are 3 × 6 = 18 cm and 2 × 6 = 12 cm.

5. Perimeter of the rectangle = 2 × (length + breadth) = 2 × (18 + 12) = 2 × 30 = 60 cm.

Cross-check: Area of the rectangle with sides 18 cm and 12 cm = 18 × 12 = 216 cm2, which matches the trapezium's area found in the first step, confirming that the perimeter of 60 cm is consistent with both conditions of the question.