The diagonals of two squares are in the ratio 2:5. Find the ratio of their areas.

Introduction / Context:This question tests similarity scaling for squares. In a square, the diagonal is directly proportional to the side length (diagonal = side*sqrt(2)). That means if diagonals are in a given ratio, the side lengths are in the same ratio. Area of a square is proportional to the square of its side length (Area = side^2). Therefore, the ratio of areas equals the square of the ratio of diagonals. This is a classic “linear ratio to area ratio” conversion problem.

Given Data / Assumptions:

• Diagonal1:Diagonal2 = 2:5
• For squares, side is proportional to diagonal
• Area is proportional to (side)^2

Concept / Approach:If d1:d2 = 2:5, then s1:s2 = 2:5. Hence Area1:Area2 = s1^2:s2^2 = 2^2:5^2 = 4:25.

Step-by-Step Solution: d1:d2 = 2:5 Since d = s*sqrt(2), side ratio s1:s2 = d1:d2 = 2:5 Area ratio = s1^2:s2^2 = 2^2:5^2 Area ratio = 4:25

Verification / Alternative check:Assume diagonals are 2 and 5. Then sides are 2/sqrt(2) and 5/sqrt(2). Areas become (2^2/2)=2 and (5^2/2)=12.5. Ratio 2:12.5 simplifies to 4:25, matching the result.

Why Other Options Are Wrong: 2:5 is the diagonal ratio, not the area ratio. 3:25, 3:15, and 5:25 come from wrong squaring or partial simplification errors.

Common Pitfalls:Forgetting to square the ratio for areas, mixing up diagonal ratio with side ratio, or simplifying incorrectly after squaring.

Final Answer:The ratio of their areas is 4:25.