Question

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your answer is incorrect.
a triangular prism and its net are shown below.
the top and bottom of the prism are shaded.
(all lengths are in centimeters.)
(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

• For \(A\): It is the length of one leg of the right - triangle. From the triangular face of the prism with sides \(3\), \(4\), \(5\) (by Pythagorean theorem \(3^{2}+4^{2}=9 + 16=25=5^{2}\)), \(A = 4\) cm.
• For \(B\): It is the height (length) of the prism. \(B=10\) cm.
• For \(C\): It is the length of one leg of the right - triangle. \(C = 3\) cm.
• For \(D\): It is the hypotenuse of the right - triangle. \(D = 5\) cm.

Step2: Calculate lateral surface area

The lateral surface area of a triangular prism \(S_{l}\) (sum of the areas of the rectangular faces). The formula for the lateral surface area of a triangular prism is \(S_{l}=(a + b + c)h\), where \(a\), \(b\), \(c\) are the sides of the triangular base and \(h\) is the height (length) of the prism. Here \(a = 3\), \(b = 4\), \(c = 5\) and \(h = 10\).

$$S_{l}=(3 + 4+5)\times10=(12)\times10 = 120\space cm^{2}$$

Step3: Calculate total surface area

The total surface area of a triangular prism \(S_{t}=S_{l}+2\times S_{\triangle}\). The area of a right - triangle \(S_{\triangle}=\frac{1}{2}\times base\times height\). Here \(base = 3\) and \(height = 4\), so \(S_{\triangle}=\frac{1}{2}\times3\times4=6\space cm^{2}\)

$$S_{t}=120+2\times6=120 + 12=132\space cm^{2}$$

Answer:

a) \(A = 4\) cm, \(B = 10\) cm, \(C = 3\) cm, \(D = 5\) cm
b) \(120\space cm^{2}\)
c) \(132\space cm^{2}\)