Question
The Pythagorean Theorem in Three Dimensions Practice
Complete this assessment to review what you've learned. It will not count toward your grade.
3 of 53 of 5 Items
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.
Remaining Attempts : 2
Complete this assessment to review what you've learned. It will not count toward your grade.
3 of 53 of 5 Items
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.
Remaining Attempts : 2
Answers
GPT-4o mini
To determine the height \( h \) of the square pyramid, we can use the relationship between the height, the slant height \( s \), and the radius \( r \) (which is half the length of a base edge of the pyramid).
1. **Understanding the Geometry of the Pyramid**:
- The lateral edge length \( e \) of the pyramid is 25 in.
- The slant height \( s \) is 24 in.
- The radius \( r \) can be calculated as half of the base length. For a square pyramid, the radius \( r \) is also the distance from the apex of the pyramid to the midpoint of one of the base edges.
2. **Finding the Length of \( r \)**:
In a right triangle formed by the height \( h \), the radius \( r \), and the slant height \( s \),
\[
s^2 = h^2 + r^2
\]
We can solve for \( h \) using \( r \) once we find the base edge length.
3. **Using the Lateral Edge**:
We have an additional triangle formed by the lateral edge \( e \), the height \( h \), and the radius \( r \):
\[
e^2 = h^2 + r^2
\]
With \( e = 25 \text{ in} \),
\[
25^2 = h^2 + r^2 \implies 625 = h^2 + r^2
\]
4. **From the Slant Height**:
We also know:
\[
s^2 = h^2 + r^2 \quad \text{with } s = 24 \text{ in}.
\]
Therefore,
\[
24^2 = h^2 + r^2 \implies 576 = h^2 + r^2
\]
5. **Equating the Two Equations**:
From the two equations:
\[
625 = h^2 + r^2 \quad \text{(1)}
\]
\[
576 = h^2 + r^2 \quad \text{(2)}
\]
6. **Finding \( h \)**:
Set both equations equal:
\[
625 = 576 + (h^2 + r^2) \implies 625 - 576 = 0 \implies h^2 = 625 - 576 = 49
\]
Then, take the square root:
\[
h = \sqrt{49} = 7 \text{ in}.
\]
Thus, the height of the square pyramid is **7 inches**.
1. **Understanding the Geometry of the Pyramid**:
- The lateral edge length \( e \) of the pyramid is 25 in.
- The slant height \( s \) is 24 in.
- The radius \( r \) can be calculated as half of the base length. For a square pyramid, the radius \( r \) is also the distance from the apex of the pyramid to the midpoint of one of the base edges.
2. **Finding the Length of \( r \)**:
In a right triangle formed by the height \( h \), the radius \( r \), and the slant height \( s \),
\[
s^2 = h^2 + r^2
\]
We can solve for \( h \) using \( r \) once we find the base edge length.
3. **Using the Lateral Edge**:
We have an additional triangle formed by the lateral edge \( e \), the height \( h \), and the radius \( r \):
\[
e^2 = h^2 + r^2
\]
With \( e = 25 \text{ in} \),
\[
25^2 = h^2 + r^2 \implies 625 = h^2 + r^2
\]
4. **From the Slant Height**:
We also know:
\[
s^2 = h^2 + r^2 \quad \text{with } s = 24 \text{ in}.
\]
Therefore,
\[
24^2 = h^2 + r^2 \implies 576 = h^2 + r^2
\]
5. **Equating the Two Equations**:
From the two equations:
\[
625 = h^2 + r^2 \quad \text{(1)}
\]
\[
576 = h^2 + r^2 \quad \text{(2)}
\]
6. **Finding \( h \)**:
Set both equations equal:
\[
625 = 576 + (h^2 + r^2) \implies 625 - 576 = 0 \implies h^2 = 625 - 576 = 49
\]
Then, take the square root:
\[
h = \sqrt{49} = 7 \text{ in}.
\]
Thus, the height of the square pyramid is **7 inches**.