Question

Triangles Unit Test
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Question
Use the image to answer the question.

An illustration shows a square pyramid with labeled parts. The front and left faces of the pyramid and base are visible. The back and right faces are not visible. The back and right base edges are not visible but their position is indicated by a dashed line. The left base edge shows a length of 16 feet. A dotted line extends from the interior of the vertex straight down to the center of the base and shows a height of 15. A right angle is indicated where the dotted height line meets the base.

Find the slant height of the square pyramid.

(1 point)
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Answer 1

To find the slant height of the square pyramid, we first need to identify the necessary dimensions from the information given.

1. The base edge of the pyramid (which is a square) is given as 16 feet.
2. The height from the vertex of the pyramid to the center of the base is given as 15 feet.

The slant height (\(l\)) of the pyramid can be found using the Pythagorean theorem. The slant height is the length of the line segment from the vertex of the pyramid to the midpoint of one of the base edges.

### Step 1: Calculate Half the Length of the Base Edge
Since the base is a square with a length of 16 feet, half of this length is:
\[
\frac{16}{2} = 8 \text{ feet}
\]

### Step 2: Use the Pythagorean Theorem
You can visualize a right triangle formed by:
- The height of the pyramid (15 feet)
- Half the base edge (8 feet)
- The slant height as the hypotenuse (\(l\))

According to the Pythagorean theorem:
\[
l^2 = height^2 + \left(\frac{base \, edge}{2}\right)^2
\]
Substituting in the values:
\[
l^2 = 15^2 + 8^2
\]
\[
l^2 = 225 + 64
\]
\[
l^2 = 289
\]
Taking the square root of both sides gives:
\[
l = \sqrt{289} = 17
\]

### Conclusion
The slant height of the square pyramid is **17 feet**.