Squares with Diagonals in Ratio — Compare Areas: The ratio concerns two squares where the diagonal of one is double the diagonal of the other. What is the ratio of their areas (larger : smaller)?

Introduction / Context:For squares, both side and diagonal are linear measures. Because area scales with the square of a linear measure, doubling a linear dimension (like the diagonal) multiplies area by the square of that factor. This is a quick test of proportional reasoning rather than computation-heavy arithmetic.

Given Data / Assumptions:

• Square 1 diagonal d1
• Square 2 diagonal d2 = 2 * d1
• Area proportionality: A ∝ (diagonal)^2 for squares

Concept / Approach:Since A ∝ d^2 for a square (because s = d/√2 and A = s^2 = d^2/2), the ratio of areas equals the ratio of squared diagonals. With d2 = 2d1, A2/A1 = (2d1)^2 / d1^2 = 4. Therefore, the larger-to-smaller area ratio is 4:1.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 2:1 assumes linear, not quadratic, scaling.
• 3:1 and 3:2 arise from ad-hoc, unsupported factors.
• 5:2 ≈ 2.5:1 is also inconsistent with quadratic scaling.

Common Pitfalls:

• Confusing side or diagonal doubling with doubling area; area quadruples when a linear dimension doubles.

Final Answer:4:1.