Question

A right triangular prism and its net are shown below.(All lengths are in millimeters.)
8
15
17
4
A
B
C
D

(a) Find the following side lengths for the net.
=A
mm
=B
mm
=C
mm
=D
mm
(b) Use the net to find the surface area of the prism.
mm2

Answer 1

To find the side lengths A, B, C, and D of the net of the right triangular prism as well as the surface area, let's start with the information provided:

Given that the dimensions involved are:
- The base of the right triangle = 8 mm
- The height of the right triangle = 15 mm
- The hypotenuse of the right triangle = 17 mm

### Part (a): Finding Side Lengths

In a right triangular prism, the net consists of two right triangles and three rectangular faces. Let's denote:
- Triangle sides:
- A: One leg of the triangle (height) = 15 mm
- B: The other leg of the triangle (base) = 8 mm
- C: The hypotenuse = 17 mm
- Rectangular faces:
- D: The height of the prism (the length of the prism, or how tall it is, which is not given in your question).

Since D is not provided, we can't compute its value without additional information.

So far, we have:
- A = 15 mm
- B = 8 mm
- C = 17 mm
- D is unknown.

### Part (b): Surface Area of the Prism

To find the surface area of a right triangular prism, we add the areas of the two triangular bases and the three rectangular sides.

1. **Area of the triangular bases**:
The area \( A_T \) of a triangle is given by:
\[
A_T = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 15 = 60 \, \text{mm}^2
\]
Since there are two triangles, the total area of the triangular bases is:
\[
2 \times A_T = 2 \times 60 = 120 \, \text{mm}^2
\]

2. **Area of the rectangular sides**:
The areas of the three rectangles are:
- Rectangle 1 (base 8 mm, height D mm): \( A_1 = 8 \times D \)
- Rectangle 2 (height 15 mm, height D mm): \( A_2 = 15 \times D \)
- Rectangle 3 (hypotenuse 17 mm, height D mm): \( A_3 = 17 \times D \)

The total area of the rectangular sides is:
\[
A_R = A_1 + A_2 + A_3 = 8D + 15D + 17D = 40D \, \text{mm}^2
\]

3. **Total Surface Area**:
The total surface area \( SA \) of the prism is:
\[
SA = \text{Area of triangular bases} + \text{Area of rectangular sides} = 120 + 40D
\]
To find the surface area, we need the height \( D \) of the prism.

If you have the value of \( D \), substitute it in the equation to find the total surface area.

If you provide the height \( D \) or if you have any other questions or need further clarification, please let me know!