To determine the ratio of the area of two similar triangles when given the ratio of their perimeters, we can use the properties of similar figures.
For similar triangles (or any similar geometric figures), the ratio of their perimeters is equal to the ratio of their corresponding side lengths. If the ratio of the perimeters is given as 9:16, we can express this as:
Ratio of side lengths = 9:16
When it comes to finding the ratio of the areas of similar triangles, the areas are proportional to the square of the ratio of their corresponding side lengths. Therefore, we can calculate the ratio of the areas as follows:
Ratio of areas = (Ratio of side lengths)² = (9:16)²
This translates to:
Ratio of areas = 9² : 16²
Calculating the squares, we get:
Ratio of areas = 81 : 256
So, the ratio of the area of the two similar triangles is 81:256.