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The Pythagorean Theorem in Three Dimensions Quick Check

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Question
Use the image to answer the question.

An illustration shows a square pyramid with labeled parts. The left, front and right faces of the pyramid and base are visible. The back face is not visible. The back base edge is not visible but its position is indicated by a dashed line. The front base edge shows a length of 40 feet. A dotted line extends from the interior of the vertex straight down to the center of the base and is labeled h. A right angle is indicated where the dotted height line meets the base. Another dotted line extends from the center of the base where the height line meets to the center of the right base edge. A dotted line extends from the right base edge straight up the center of the right face of the prism and meets the height line at the vertex. This dotted line shows a slant height of 25 feet.

Apply the Pythagorean Theorem to determine the height of the square pyramid, h
. Round the answer to the nearest tenth.

(1 point)
Responses

32.0 ft.
32.0 ft.

31.2 ft.
31.2 ft.

47.2 ft.
47.2 ft.

15.0 ft.
15.0 ft.
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1 answer

To find the height \( h \) of the square pyramid using the Pythagorean Theorem in three dimensions, we can set up the necessary relationships between the height, the slant height, and half the length of the base.

Let's break down the problem:

  1. From the description, we know:

    • The length of the base edge is 40 feet. Therefore, half of the base (which we need for our calculations) is: \[ \frac{40}{2} = 20 \text{ feet} \]

  2. The slant height \( s \) is given as 25 feet.

  3. The relationship we have is a right triangle formed by the height \( h \), half of the base (20 feet), and the slant height (25 feet). Using the Pythagorean Theorem: \[ s^2 = h^2 + \left(\frac{base}{2}\right)^2 \]

    Plugging in the values, we get: \[ 25^2 = h^2 + 20^2 \]

    This simplifies to: \[ 625 = h^2 + 400 \]

  4. Subtract 400 from both sides: \[ h^2 = 625 - 400 \] \[ h^2 = 225 \]

  5. Taking the square root of both sides, we find: \[ h = \sqrt{225} = 15 \text{ feet} \]

So the height \( h \) of the square pyramid is 15.0 ft. The correct response is:

15.0 ft.