Question

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=square mathrm{cm} ) ( b=square mathrm{cm} ) ( c=square mathrm{cm} ) ( d=square mathrm{cm} ) (b) use the net to find the lateral surface area of the prism. neither the top nor bottom is included. ( square mathrm{cm}^{2} ) (c) use the net to find the total surface area of the prism. ( square mathrm{cm}^{2} )

Explanation:

Part (a)

Step1: Identify side lengths from the prism

• The triangular prism has rectangles in its lateral faces. By comparing the prism and the net:
• \(A = 9\) cm (height of the prism, corresponding to the vertical side of the rectangles).
• \(B = 13\) cm (hypotenuse of the triangular base, which is one of the lengths of the rectangles).
• \(C = 12\) cm (one of the legs of the triangular base, corresponding to a side of the rectangle).
• \(D = 5\) cm (the other leg of the triangular base, corresponding to a side of the rectangle).

Part (b)

Step1: Recall the formula for lateral - surface area of a triangular prism

The lateral - surface area \(LSA\) of a triangular prism is \(LSA=(a + b+ c)h\), where \(a\), \(b\), \(c\) are the sides of the triangular base and \(h\) is the height of the prism. Here, \(a = 5\), \(b = 12\), \(c = 13\) and \(h = 9\).

$$LSA=(5 + 12+13)\times9$$

Step2: Simplify the expression

First, \(5 + 12+13=30\). Then \(30\times9 = 270\) \(cm^{2}\).

Part (c)

Step1: Recall the formula for the area of a right - triangle

The area of a right - triangle \(A=\frac{1}{2}\times base\times height\). For the triangular base with base \(b = 5\) and height \(h = 12\), \(A_{triangle}=\frac{1}{2}\times5\times12\).

$$A_{triangle}=\frac{1}{2}\times5\times12=30$$

\(cm^{2}\)

Step2: Recall the formula for the total surface area of a triangular prism

The total surface area \(TSA = LSA+2A_{triangle}\). We know \(LSA = 270\) \(cm^{2}\) and \(A_{triangle}=30\) \(cm^{2}\).

$$TSA=270 + 2\times30$$

Step3: Simplify the expression

$$TSA=270+60=330$$

\(cm^{2}\)

Answer:

(a) \(A = 9\) cm, \(B = 13\) cm, \(C = 12\) cm, \(D = 5\) cm.
(b) \(270\) \(cm^{2}\)
(c) \(330\) \(cm^{2}\)