The similarity ratio of two similar polygons is 2 to 3. Comparing the smaller polygon to the larger polygon, find the ratio of their areas. Choose from 2:3, 3:2, 9:4, or 4:9.

Step-by-step solution

1. 1

   Separate the two different ratios in play. A similarity ratio is a ratio of lengths. The question asks for a ratio of areas, and those two ratios are never equal unless the figures are congruent. Picking $2:3$ straight off the page is the standard trap here.

2. 2

   See where the square comes from. Split each polygon into triangles. Every triangle in the larger figure has base and height both $\tfrac{3}{2}$ times the matching triangle in the smaller one, so its area $\tfrac12 bh$ is multiplied twice:

   $\frac{\text{area}_{\text{large}}}{\text{area}_{\text{small}}}=\frac{\tfrac12\left(\tfrac32 b\right)\left(\tfrac32 h\right)}{\tfrac12 bh}=\left(\frac{3}{2}\right)^{2}$

   Adding the triangles back up keeps that same factor, so the rule holds for the whole polygon.

3. 3

   Apply the rule in the order the question asks. Smaller to larger means the side ratio is $\dfrac{2}{3}$, so square it:

   $\frac{A_{\text{small}}}{A_{\text{large}}}=\left(\frac{2}{3}\right)^{2}=\frac{2^2}{3^2}=\frac{4}{9}$

4. 4

   Check the answer against the distractors. $2:3$ forgets to square, $3:2$ and $9:4$ both reverse the order and describe the larger figure compared to the smaller. Only $4:9$ is both squared and in the right order.

5. 5

   State the result. The ratio of the areas, smaller to larger, is

   $A_{\text{small}}:A_{\text{large}}=4:9$