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: Identify side - lengths from the prism and net

By observing the 3 - D prism and its net, we can match the sides. The height of the prism gives $A = 11$ cm, the hypotenuse of the triangular face gives $B = 10$ cm, one of the legs of the triangular face gives $C = 6$ cm, and the other leg of the triangular face gives $D = 8$ cm.

Step2: Calculate the lateral surface area

The lateral surface area of a triangular prism (excluding top and bottom) is the sum of the areas of the rectangular sides. The rectangular sides have dimensions: $11\times8$, $11\times6$, and $11\times10$. The lateral surface area $S_{lateral}=11\times8 + 11\times6+11\times10=11\times(8 + 6+10)=11\times24 = 264$ $cm^{2}$.

Step3: Calculate the total surface area

First, find the area of the triangular faces. The area of a right - triangle with legs $a = 6$ cm and $b = 8$ cm is $A_{triangle}=\frac{1}{2}\times6\times8 = 24$ $cm^{2}$. The total surface area $S_{total}=S_{lateral}+2\times A_{triangle}=264+2\times24=264 + 48=312$ $cm^{2}$.

Answer:

(a) $A = 11$ cm, $B = 10$ cm, $C = 6$ cm, $D = 8$ cm
(b) $264$ $cm^{2}$
(c) $312$ $cm^{2}$