Coordinate Geometry and Nets Unit Test
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Question
Use the image to answer the question.
An illustration shows the net of a triangular prism. The illustration forms a shape where the small triangle is in the middle with each side being the base of three different triangles. The height of the triangle in the middle is 5 inches as shown by a dashed line and the base is 6 inches.
Use the net to find the slant height of the triangular pyramid made of equilateral triangles if its surface area is 78 in.2 .
(1 point)
inches
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To find the slant height of the triangular pyramid, we first need to find the slant height of one of the triangles in the net.
Each of the equilateral triangles in the net has a base of 6 inches and a height of 5 inches.
To find the slant height of the equilateral triangle, we can use the Pythagorean theorem:
\( \text{slant height} = \sqrt{(\text{height})^2 + (\frac{1}{2} \times \text{base})^2} \)
\( \text{slant height} = \sqrt{(5)^2 + (\frac{1}{2} \times 6)^2} \)
\( \text{slant height} = \sqrt{25 + 9} \)
\( \text{slant height} = \sqrt{34} \)
Therefore, the slant height of each triangular face is approximately 5.83 inches.
Since the surface area of the triangular pyramid is 78 in^2 and there are 3 equilateral triangles making up the surface area of the pyramid, the total surface area of the 3 equilateral triangles is 78 in^2.
\( \text{Total surface area of the 3 equilateral triangles} = 78 in^2 \)
\( \text{Surface area of one equilateral triangle} = \frac{78}{3} = 26 in^2 \)
Let the side length of the equilateral triangle be \( s \).
The formula for the surface area of an equilateral triangle with side length \( s \) is \( \frac{\sqrt{3}}{4} \times s^2 \). Setting this equal to 26, we get:
\( \frac{\sqrt{3}}{4} \times s^2 = 26 \)
\( s^2 = \frac{26 \times 4}{\sqrt{3}} \)
\( s^2 = \frac{104}{\sqrt{3}} \)
Now, the slant height of the pyramid is the height of the equilateral triangle:
\( \text{slant height of pyramid} = \sqrt{(\text{height})^2 + (\frac{1}{2} \times s)^2} \)
Substitute the value of \( s \) for the height and calculate.