The Pythagorean Theorem in Three Dimensions Practice
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, but its position is indicated by a dashed line. The back base edge is not visible, but its position is indicated by a dashed line. The left base edge is labeled a. The back left lateral edge is labeled e. 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 and is labeled r. 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. The dotted slant height line is labeled s.
For the square pyramid, the lateral edge length, e, is 25 in., and the slant height, s, is 24 in. Find the height of the pyramid, h. Round the answer to the nearest whole number. Hint: Before finding the length of h, you will first need to find the length of r
.
(1 point)
The height of the square pyramid is
in.
To find the height of the pyramid, we can use the Pythagorean Theorem in three dimensions.
First, let's find the length of r. We know that r is the distance from the center of the base to the center of the right base edge. Since the base of the pyramid is a square, the distance from the center of the base to the right base edge is half the length of the base. The lateral edge length, a, is labeled as 25 inches. So, the length of the base would be 2a = 2(25) = 50 inches. Therefore, r = 50/2 = 25 inches.
Next, let's find the length of h. Since h is the height of the pyramid, it is the perpendicular distance from the apex (vertex) to the base. We can use the Pythagorean Theorem to find h.
Using the right triangle formed by h, r, and s, we have:
h^2 + r^2 = s^2
Substituting the values we have:
h^2 + (25)^2 = (24)^2
h^2 + 625 = 576
h^2 = 576 - 625
h^2 = -49
Since the square of a real number cannot be negative, it means that there is no real solution for h in this case. Therefore, it is not possible to determine the height of the pyramid based on the given information.