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? \bigcirc the surface areas are the same. \bigcirc the surface area of the larger prism is 2 times the surface area of the smaller prism. \bigcirc the surface area of the larger prism is 4 times the surface area of the smaller prism. \bigcirc the surface area of the larger prism is 8 times the surface area of the smaller prism.

Explanation:

Step1: Recall the property of similar solids

For similar solids, if the scale factor (ratio of corresponding linear measurements) is \( k \), the ratio of their surface areas is \( k^2 \). Here, the perimeter of each face of one prism is double the perimeter of the corresponding face of the other. Since perimeter is a linear measurement, the scale factor \( k \) between the two similar triangular prisms is \( 2 \) (larger to smaller or vice - versa, but we are interested in the ratio of surface areas).

Step2: Calculate the ratio of surface areas

Using the formula for the ratio of surface areas of similar solids \( \text{Ratio of surface areas}=k^2 \), where \( k = 2 \). So, \( \text{Ratio of surface areas}=2^2=4 \). This means that if the scale factor of the linear dimensions (like perimeter of faces) is 2, the ratio of the surface area of the larger prism to the smaller prism is 4. So the surface area of the larger prism is 4 times the surface area of the smaller prism.

Answer:

The surface area of the larger prism is 4 times the surface area of the smaller prism. (Corresponding to the option: The surface area of the larger prism is 4 times the surface area of the smaller prism.)