The length and breadth of a rectangle add up to 6 cm. A square is constructed such that each side of the square 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 of the square (in cm²)?

Introduction:
This is a geometry and algebra problem involving a rectangle and a square whose side equals the rectangle diagonal. You are given the sum of rectangle sides and a ratio of areas, and asked to find the area of the square. The key idea is to express the diagonal and areas in terms of the rectangle sides and then use the given ratio to solve for the areas.

Given Data / Assumptions:

• Let the rectangle have length l and breadth b.
• l + b = 6 cm.
• Diagonal of rectangle d = √(l² + b²).
• A square is constructed whose side is equal to d.
• Area of rectangle = l * b.
• Area of square = d² = l² + b².
• Ratio of areas (square : rectangle) = 5 : 2.
• We must find the area of the square in cm².

Concept / Approach:
Use algebraic identities. Since area of square is l² + b² and area of rectangle is l * b, the ratio condition becomes (l² + b²)/(l * b) = 5/2. Also, from l + b = 6 we can relate l² + b² to l * b using the identity (l + b)² = l² + 2lb + b². From these two relations, we can solve for l * b, then find l² + b² and hence the square area.

Step-by-Step Solution:
Step 1: Express l² + b² in terms of l + b and l * b.(l + b)² = l² + 2lb + b².So l² + b² = (l + b)² − 2lb.Given l + b = 6, so (l + b)² = 36.Hence l² + b² = 36 − 2lb.Step 2: Use the area ratio condition.(l² + b²)/(l * b) = 5/2.Substitute l² + b²:(36 − 2lb) / (lb) = 5/2.Step 3: Solve for lb.36 − 2lb = (5/2)lb.Multiply both sides by 2: 72 − 4lb = 5lb ⇒ 72 = 9lb ⇒ lb = 8.Step 4: Find the square area.Area of square = l² + b² = 36 − 2lb = 36 − 2 * 8 = 36 − 16 = 20 cm².

Verification / Alternative check:
We can find l and b explicitly. From l + b = 6 and lb = 8, l and b are roots of t² − 6t + 8 = 0. Solving: t = [6 ± √(36 − 32)]/2 = (6 ± 2)/2 ⇒ t = 4 or 2. Thus rectangle sides are 4 cm and 2 cm. Diagonal d = √(4² + 2²) = √(16 + 4) = √20, so the square side is √20 and its area is (√20)² = 20 cm², confirming the earlier result.

Why Other Options Are Wrong:
10 cm² and 25 cm² are inconsistent with the given ratio. Options involving roots like 4√5 cm² or 10√2 cm² look attractive but represent lengths or misinterpreted areas rather than the correct area value. Only 20 cm² satisfies both the side-sum condition and the 5 : 2 area ratio.

Common Pitfalls:
Learners sometimes confuse side length of the square with its area, or they directly assume specific values for l and b without using the ratio condition properly. Another frequent error is to forget the identity l² + b² = (l + b)² − 2lb and instead try to work with three unknowns. Using the identities smartly simplifies the algebra considerably.

Final Answer:
The area of the square is 20 cm².