To find the area of the triangular base of the prism, we first need to determine the dimensions of the triangular base using the information given.
The net consists of two right triangles placed above and below the middle rectangle, which has an area of 64 square inches. The height of these right triangles (the perpendicular side) is given as 6 inches. We need to find the base of these triangles.
From the area formula for a triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Let \( b \) be the base of one of the triangles. Given the height (6 inches), we can express the area of one triangular base as follows: \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times b \times 6 \]
To find the base \( b \), we can set the area of the triangle equal to the area of the triangular base. Since there are two identical triangles in the prism's net: \[ \text{Total Area}_{\text{triangles}} = 2 \times \frac{1}{2} \times b \times 6 = 6b \]
Now, we know the area of the prism consists of the three rectangles plus the areas of the two triangles. The total area of the triangular bases (composed of both triangles) must balance with the combined areas of the rectangles. The total rectangle area is: \[ 48 + 64 + 80 = 192 \text{ square inches} \]
If the total area of the triangular bases is to complement the area of the rectangles: \[ 6b + 192 = \text{Total Area} \]
The base \( b \) corresponds to the question mark in the illustration.
From the measurements, we can solve for \( b \):
- Each triangle's area (as discussed) should correspond to how they complete the rectangular areas.
- We calculate that the two triangles together contribute \( 6b \).
In this specific question, we need just the area of one triangular base.
Assuming \( b = 8 \): \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times 8 \times 6 = 24 \text{ square inches} \]
Hence, the area of the triangular base of the prism is: \[ \boxed{24 \text{ in.}^2} \]