Question

two triangular prisms are similar. the perimeter of each face of one prism is double the perimeter of the corresponding face of the other prism. how are the surface areas of the figures related? the surface areas are the same. the surface area of the larger prism is 2 times the surface area of the smaller prism. the surface area of the larger prism is 4 times the surface area of the smaller prism. the surface area of the larger prism is 8 times the surface area of the smaller prism.

Explanation:

Step1: Recall the ratio - area relationship for similar figures

For two similar figures, if the ratio of their corresponding linear - dimensions (such as perimeters) is \(k\), the ratio of their surface areas is \(k^{2}\).
Let the perimeter of the smaller prism's face be \(P_1\) and the perimeter of the larger prism's corresponding face be \(P_2\). Given that \(P_2 = 2P_1\), so the scale factor \(k=\frac{P_2}{P_1}=2\).

Step2: Calculate the ratio of the surface areas

The ratio of the surface area of the larger prism \(S_2\) to the surface area of the smaller prism \(S_1\) is \(k^{2}\). Substituting \(k = 2\) into the formula \(\frac{S_2}{S_1}=k^{2}\), we get \(\frac{S_2}{S_1}=2^{2}=4\), which means \(S_2 = 4S_1\).

Answer:

The surface area of the larger prism is 4 times the surface area of the smaller prism.