To solve the problem, we need to extract dimensions from the right triangular prism and its net.
(a) Finding side lengths for the net:
For a right triangular prism:
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Identify the base triangle. Given the side lengths of 12 mm, 5 mm, and 13 mm, it's important to check if these lengths form a right triangle. The hypotenuse (the longest side) is 13 mm.
To verify, we can use the Pythagorean theorem: \[ 12^2 + 5^2 = 144 + 25 = 169 \] \[ 13^2 = 169 \] Thus, sides 12 mm, 5 mm, and 13 mm indeed form a right triangle.
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Label the sides:
- We'll designate the sides of the triangle as follows:
- \( A = 5 , \text{mm} \) (one leg of the triangle)
- \( B = 12 , \text{mm} \) (the other leg of the triangle)
- \( C = 13 , \text{mm} \) (the hypotenuse)
For a right triangular prism, the other dimensions related to the height of the prism (the length parallel to the hypotenuse) should be specified but are not given, thus we only need to ascertain the side lengths mentioned. The label for side \( D \) isn't explicitly defined from given information, but it would typically represent the height of the prism.
- We'll designate the sides of the triangle as follows:
Assuming the height of the prism is not known from the problem details, let's denote it as height \( h \). Common representations would suggest this could be known from the context or diagram.
Thus, taking standard practices:
- \( A = 5 , \text{mm} \)
- \( B = 12 , \text{mm} \)
- \( C = 13 , \text{mm} \)
- \( D = h , \text{mm} \) (height - unknown)
(b) Finding the surface area of the prism:
The surface area \( S \) of a triangular prism can be formulated as:
\[ S = \text{Area of base} + \text{Lateral surface area} \]
- Find the area of the base triangle:
Using the formula for the area of a right triangle: \[ \text{Area of base} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 12 = 30 , \text{mm}^2 \]
- Calculate the lateral surface area:
The lateral surface area comprises three rectangles:
- 2 rectangles corresponding to the two sides of the triangle (the legs):
- Rectangle with side \( A = 5 , \text{mm} \) and height \( h \): \( 2 \times (h \times A = h \times 5 = 10h) \)
- Rectangle with side \( B = 12 , \text{mm} \) and height \( h \): \( 2 \times (h \times B = h \times 12 = 24h) \)
- 1 rectangle corresponding to the hypotenuse \( C = 13 , \text{mm} \) and height \( h \): \( (h \times C = h \times 13 = 13h) \)
Thus, the lateral surface area becomes: \[ \text{Lateral area} = (5h + 12h + 13h) = 30h \]
- Total surface area: \[ S = \text{Area of base} + \text{lateral surface area} = 30 + 30h \]
If we had the height \( h \), we could find a specific area. Without it, the surface area will be given as: \[ S = 30 + 30h , \text{mm}^2 \]
Unless \( h \) is specified, we conclude with the above general total surface area expression.