Question

Use the image to answer the question.

An illustration shows the net of a triangular prism. The illustration forms a shape where the small triangle is in the middle with each side being the base of three different triangles. The height of the triangle in the middle is 5 inches as shown by a dashed line and the base is 6 inches.

Use the net to find the slant height of the triangular pyramid made of equilateral triangles if its surface area is 78 in.2 .

(1 point)
inches
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Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the slant height of the triangular pyramid, we first need to find the lateral area of the triangular prism, which is half of the surface area of the triangular pyramid.

The lateral area of the triangular prism can be calculated by finding the area of the three triangles surrounding the middle triangle. Each of these triangles is half of an equilateral triangle with a base of 6 inches and a height of 5 inches.

Area of each triangle = (1/2) * base * height
Area of each triangle = (1/2) * 6 * 5
Area of each triangle = 15 in^2

Since there are three identical triangles, the total lateral area of the triangular prism is 3 * 15 = 45 in^2.

Since the surface area of the triangular pyramid is 78 in^2, the lateral area of the triangular prism is half of that, which is 39 in^2.

Now, we need to find the slant height of the triangular pyramid, which is the height of the middle triangle in the net of the triangular prism. From the net, we can see that the height of the middle triangle is 5 inches.

Therefore, the slant height of the triangular pyramid is 5 inches.