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

Introduction / Context:
This question tests scaling laws for similar shapes. Squares are always similar, so any linear measurement (like side, diagonal, perimeter) scales by the same factor between two squares. The diagonal of a square is proportional to its side because diagonal = side*sqrt(2). Therefore, if diagonals are in the ratio 2:5, the sides are also in the ratio 2:5. Area depends on the square of a linear dimension, so the ratio of areas becomes the square of the ratio of sides (or diagonals). This is a common aptitude shortcut: linear ratio k leads to area ratio k^2.

Given Data / Assumptions:

• Diagonal ratio d1:d2 = 2:5
• For a square, diagonal is proportional to side
• Area of a square is proportional to side^2

Concept / Approach:
Convert diagonal ratio to side ratio (same), then square it to get area ratio: 2^2:5^2 = 4:25.

Step-by-Step Solution:
d1:d2 = 2:5 Since d = s*sqrt(2), 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) and (5^2/2), i.e., 2 and 12.5. Ratio 2:12.5 simplifies to 4:25, confirming the same result.

Why Other Options Are Wrong:
2:25 and 10:25 come from incorrect squaring or multiplying ratios. 3:25 and 6:25 are random-looking distractors that do not match squaring 2:5.

Common Pitfalls:
Using the diagonal ratio directly as the area ratio (not squaring), or mixing up which ratio to square.

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