Discussion

If ratio of the areas of two squares is 1:4, the ratio of their perimeters is

Answer: A

Area varies as the square of the side, so take square roots to recover the length ratio.

\[ \frac{a_{1}^{2}}{a_{2}^{2}} = \frac{1}{4} \quad\Longrightarrow\quad \frac{a_{1}}{a_{2}} = \frac{1}{2} \]

Perimeter is \( 4a \), a plain multiple of the side, so it keeps that same ratio:

\[ \frac{P_{1}}{P_{2}} = \frac{4a_{1}}{4a_{2}} = \frac{1}{2} \quad\Longrightarrow\quad 1 : 2 \]

Carrying the area ratio straight through to the perimeters gives 1 : 4, the first distractor.

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