If the ratio of the areas of two squares is 9 : 1, find the ratio of their perimeters.

Introduction / Context:
For similar figures (all squares are similar), linear measures scale as the square root of area ratios. Thus, side ratio equals the square root of the area ratio, and perimeter, being linear in side, shares the same ratio as sides.

Given Data / Assumptions:

• Area ratio A1 : A2 = 9 : 1
• Squares ⇒ same shape, so side ratio = sqrt(area ratio)

Concept / Approach:
If s1 and s2 are sides, then s1^2 : s2^2 = 9 : 1 ⇒ s1 : s2 = 3 : 1. Perimeter P = 4s, hence P1 : P2 = s1 : s2 = 3 : 1.

Step-by-Step Solution:
s1 : s2 = sqrt(9 : 1) = 3 : 1P1 : P2 = 4s1 : 4s2 = s1 : s2 = 3 : 1

Verification / Alternative check:
Take sample squares with sides 3 and 1: areas 9 and 1 (ratio 9:1); perimeters 12 and 4 (ratio 3:1) — consistent.

Why Other Options Are Wrong:
9:1 is the area ratio, not the perimeter ratio; 3:4 and 1:3 contradict the square-root relation for linear dimensions.

Common Pitfalls:
Carrying the area ratio directly over to perimeters; always switch to linear scaling via square root for similar shapes.

Final Answer:
3:1