The sum of the length and breadth of a rectangle is 6 cm. A square is constructed whose side is equal to the diagonal of this rectangle. If the ratio of the area of the square to the area of the rectangle is 5 : 2, what is the area (in cm²) of the square?

Introduction / Context:
This question links a rectangle and a square through the diagonal of the rectangle. The diagonal is used as the side length of the square. Additionally, a ratio of areas between the square and the rectangle is given. The problem involves using the relationships between the sides of a rectangle, its diagonal, and the area expression, along with the area ratio, to determine the area of the square. This tests algebraic manipulation and geometric understanding of rectangles and squares.

Given Data / Assumptions:

• Let the rectangle have length l and breadth b.
• Sum of length and breadth: l + b = 6 cm.
• Diagonal of rectangle = d = √(l^2 + b^2).
• A square is formed with side equal to d, so area of square = d^2.
• Ratio of area of the square to the area of the rectangle is 5 : 2.
• Area of rectangle = l * b.

Concept / Approach:
The area of the square formed on the diagonal is d^2, and for the rectangle, the area is l * b. Given that the ratio of the areas is 5 : 2, we have d^2 / (l * b) = 5 / 2. But d^2 = l^2 + b^2. Additionally, (l + b)^2 = l^2 + b^2 + 2lb; and we know l + b = 6, which gives an equation linking l^2 + b^2 and lb. Using these two relationships, we solve for lb and then compute the area of the square using the ratio 5 : 2.

Step-by-Step Solution:
Let l and b be the sides of the rectangle.Given l + b = 6 cm.Area of rectangle = l * b.Diagonal d satisfies d^2 = l^2 + b^2.Given area ratio: (area of square) / (area of rectangle) = 5 / 2.Area of square = d^2 = l^2 + b^2.So (l^2 + b^2) / (l * b) = 5 / 2.This implies 2(l^2 + b^2) = 5 l b.Also, (l + b)^2 = l^2 + b^2 + 2 l b = 6^2 = 36.From 2(l^2 + b^2) = 5 l b, we have l^2 + b^2 = (5 / 2) l b.Substitute into (l + b)^2 equation: (5 / 2) l b + 2 l b = 36.Combine coefficients: (5 / 2 + 2) l b = (5 / 2 + 4 / 2) l b = (9 / 2) l b.Thus (9 / 2) l b = 36, so l b = 36 * (2 / 9) = 8.Area of rectangle = l b = 8 cm².Area of square : area of rectangle = 5 : 2, so area of square = (5 / 2) * 8 = 20 cm².

Verification / Alternative check:
We can check if there exist l and b satisfying l + b = 6 and l b = 8. These are roots of the quadratic t^2 − 6t + 8 = 0, so t = 2 or 4. Thus, l = 4 and b = 2 (or vice versa). Diagonal d^2 = 4^2 + 2^2 = 16 + 4 = 20, so area of square on the diagonal is indeed 20 cm². Area of rectangle is 4 * 2 = 8 cm², and 20 : 8 simplifies to 5 : 2. This confirms that the result is consistent with the given ratio and conditions.

Why Other Options Are Wrong:
The value 10 cm² would correspond to a ratio closer to 5 : 4, not 5 : 2. The value 4√5 is not even in square units and appears to be a misinterpretation of the diagonal length. The value 10√2 would also be a misinterpretation of the diagonal rather than its squared area. The value 25 cm² does not satisfy the ratio 5 : 2 when compared with a consistent rectangle area. Only 20 cm² fits both the algebraic derivation and the given ratio exactly.

Common Pitfalls:
Some students mistakenly use the diagonal directly as the answer instead of its square when asked for the area of the square. Others might misapply the relationship between (l + b)^2 and l^2 + b^2 + 2 l b or mishandle the algebra when solving for l b. Forgetting that the ratio refers to areas and not side lengths is another common source of error. Keeping track of each step and systematically using the algebraic identities avoids these mistakes.

Final Answer:
The area of the square constructed on the diagonal of the rectangle is 20 cm².