Two squares are given such that the diagonal of the larger square is exactly double the diagonal of the smaller square. What is the ratio of their areas (larger : smaller)?

Introduction / Context: This question checks proportional reasoning in geometry. For similar shapes (like squares), if a linear dimension scales by a factor k, the area scales by k^2. The diagonal of a square is a linear measure, so doubling the diagonal means all linear dimensions (including side length) also double, which makes the area become four times.

Given Data / Assumptions:

• Diagonal of larger square = 2 * (diagonal of smaller square)
• Squares are similar, so side length scales in the same ratio as diagonal
• Area of a square is proportional to (side)^2

Concept / Approach: Use scaling: if diagonal ratio is 2:1, then side ratio is also 2:1. Therefore area ratio is (2^2):(1^2) = 4:1.

Step-by-Step Solution: Let smaller square have diagonal d and side s Larger square has diagonal 2d, so its side becomes 2s (same scale factor) Smaller area = s^2 Larger area = (2s)^2 = 4s^2 Area ratio (larger:smaller) = 4s^2 : s^2 = 4:1

Verification / Alternative check: Using a concrete example: if smaller side is 5, area is 25. Larger side is 10, area is 100. Ratio 100:25 simplifies to 4:1, confirming the result.

Why Other Options Are Wrong: 2:1: would be true only for a linear measure, not for area. 2:3 and 3:1: do not match square scaling rules. 5:1: would require a scale factor sqrt(5), not 2.

Common Pitfalls: Assuming area scales the same way as diagonal (linear) instead of squared. Mixing up diagonal ratio with area ratio. Not recognizing that squares are similar shapes.

Final Answer: Required ratio of areas = 4:1