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jmc

geometry junior

Problem

Regions I, II and III are bounded by squares. The perimeter of region I is 12 units and the perimeter of region II is 24 units. What is the ratio of the area of region I to the area of region III? Express your answer as a common fraction.

problem
Solution
A side of square I has length 3, while a side of square II has length 6 (all sides have equal length). Therefore, a side of square III has length 9. Since the side length of square I is that of square III, and the ratio of their areas is the square of the ratio of their side lengths, the ratio of the area of square I to square III is . Alternately, you can just calculate the areas: square I has an area of 9, square III has an area of 81, thus, the ratio of their areas is
Final answer
\frac{1}{9}