Using a net to find the surface area of a triangular prism (a) Find the following side lengths for the net. \[ \begin{array}{l}\text { A }=15 \mathrm{yd} \\ B=2 \mathrm{yd} \\ C=8 \mathrm{yd} \\ D=17] \mathrm{yd} \\ \text { (b) Use the net to find the surface area of the prism. } \\ \square \text { yd }{ }^{2}\end{array} \]

To find the surface area of a triangular prism using its net, first visualize the net as consisting of two triangular bases and three rectangular sides. The area of each triangular base can be calculated using the formula for the area of a triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] For this problem, assuming "A" is the base and "B" is the height, the area of one triangle is: \[ \frac{1}{2} \times 15 \text{ yd} \times 2 \text{ yd} = 15 \text{ yd}^2 \] So the total area of the two triangular bases is \(2 \times 15 \text{ yd}^2 = 30 \text{ yd}^2\). Next, calculate the area of the three rectangles using their dimensions. The rectangles have the following areas: 1. Rectangle with side \(A\) and height \(D\): \[ 15 \text{ yd} \times 17 \text{ yd} = 255 \text{ yd}^2 \] 2. Rectangle with side \(B\) and height \(D\): \[ 2 \text{ yd} \times 17 \text{ yd} = 34 \text{ yd}^2 \] 3. Rectangle with side \(C\) and height \(D\): \[ 8 \text{ yd} \times 17 \text{ yd} = 136 \text{ yd}^2 \] Now add them all up: - Total from triangular bases: \(30 \text{ yd}^2\) - Total from rectangles: \(255 \text{ yd}^2 + 34 \text{ yd}^2 + 136 \text{ yd}^2 = 425 \text{ yd}^2\) Final Surface Area of the prism: \[ 30 \text{ yd}^2 + 425 \text{ yd}^2 = 455 \text{ yd}^2 \] So, the surface area of the triangular prism is \(455 \text{ yd}^2\). And voila! You've transformed some geometry into a fun exploration of prisms!