Question

(a) find the following side lengths for the net.
a =
cm
b =
cm
c =
cm
d =
cm
(b) use the net to find the lateral surface area of the prism. neither the top nor bottom is included.

cm²
(c) use the net to find the total surface area of the prism.

cm²

Explanation:

Step1: Analyze the triangular prism net

From the 3D figure (right), the triangular base has legs 3 and 4 (since it's a right triangle, hypotenuse \( C = \sqrt{3^2 + 4^2} = 5 \)). The length of the prism (B) is 10. The height of the triangle (A) can be related, but looking at the net, the rectangular faces: the side with length D should match the height of the triangle? Wait, no—let's match with the 3D figure. The triangular base has legs 3, 4, hypotenuse 5. The length of the prism (the distance between the two triangles) is 10. So:

• \( A \): The height of the triangle? Wait, no, in the net, the small rectangles: A is the height of the triangle? Wait, the right triangle has legs 3 and 4, so area is \( \frac{3\times4}{2} \). But for the net, the sides:
• \( A \): Looking at the 3D figure, the vertical side (maybe the height of the triangle? Wait, no, the triangle has legs 3 and 4, so the sides of the triangle are 3, 4, 5. So in the net, the triangular faces have legs 3 and 4, hypotenuse 5. So:
• \( A = 3 \) (one leg of the right triangle)
• \( B = 10 \) (length of the prism)
• \( C = 5 \) (hypotenuse of the right triangle)
• \( D = 4 \) (the other leg of the right triangle)

Wait, let's confirm: the 3D figure has a triangular prism with triangular base (right triangle) with legs 3 and 4, hypotenuse 5, and length 10. So the net's triangular faces have legs 3, 4, hypotenuse 5. The rectangular faces: one with length 10 and width 3 (A=3), one with length 10 and width 4 (D=4), one with length 10 and width 5 (C=5), and the length of the prism is B=10. Wait, maybe:

• \( A \): 3 (matches the leg of the triangle)
• \( B \): 10 (length of the prism)
• \( C \): 5 (hypotenuse)
• \( D \): 4 (the other leg)

Step2: Verify each length

• \( A \): From the 3D figure, the vertical leg is 3, so \( A = 3 \)
• \( B \): The length of the prism (distance between the two triangles) is 10, so \( B = 10 \)
• \( C \): Hypotenuse of the right triangle with legs 3 and 4: \( \sqrt{3^2 + 4^2} = 5 \), so \( C = 5 \)
• \( D \): The other leg of the right triangle is 4, so \( D = 4 \)

Answer:

\( A = 3 \) cm, \( B = 10 \) cm, \( C = 5 \) cm, \( D = 4 \) cm