Question
what is the base of a square pyramid with lateral edge of 25 in and slant height of 24in
Answers
GPT-4o mini
To find the base of a square pyramid when given the lateral edge and the slant height, we can use the relationship between these measurements and the dimensions of the base.
1. **Definitions**:
- Let \( s \) be the length of one side of the square base.
- The lateral edge (25 in) is the diagonal line from the apex of the pyramid to a vertex of the base.
- The slant height (24 in) is the diagonal line from the apex of the pyramid to the midpoint of one side of the base.
2. **Visualize the Triangle**:
The lateral edge, the slant height, and half the base form a right triangle. We can denote:
- The slant height as \( h_s = 24 \) in.
- Half the side of the base as \( \frac{s}{2} \).
- The lateral edge (hypotenuse) as \( L = 25 \) in.
3. **Using the Pythagorean Theorem**:
The Pythagorean theorem states that:
\[
L^2 = h_s^2 + \left(\frac{s}{2}\right)^2
\]
Plugging in the values:
\[
25^2 = 24^2 + \left(\frac{s}{2}\right)^2
\]
Which simplifies to:
\[
625 = 576 + \left(\frac{s}{2}\right)^2
\]
Rearranging gives:
\[
\left(\frac{s}{2}\right)^2 = 625 - 576
\]
\[
\left(\frac{s}{2}\right)^2 = 49
\]
4. **Solving for \( s \)**:
Taking the square root:
\[
\frac{s}{2} = 7
\]
Therefore:
\[
s = 2 \times 7 = 14 \text{ in}
\]
Thus, the side length of the base of the square pyramid is **14 inches**.
1. **Definitions**:
- Let \( s \) be the length of one side of the square base.
- The lateral edge (25 in) is the diagonal line from the apex of the pyramid to a vertex of the base.
- The slant height (24 in) is the diagonal line from the apex of the pyramid to the midpoint of one side of the base.
2. **Visualize the Triangle**:
The lateral edge, the slant height, and half the base form a right triangle. We can denote:
- The slant height as \( h_s = 24 \) in.
- Half the side of the base as \( \frac{s}{2} \).
- The lateral edge (hypotenuse) as \( L = 25 \) in.
3. **Using the Pythagorean Theorem**:
The Pythagorean theorem states that:
\[
L^2 = h_s^2 + \left(\frac{s}{2}\right)^2
\]
Plugging in the values:
\[
25^2 = 24^2 + \left(\frac{s}{2}\right)^2
\]
Which simplifies to:
\[
625 = 576 + \left(\frac{s}{2}\right)^2
\]
Rearranging gives:
\[
\left(\frac{s}{2}\right)^2 = 625 - 576
\]
\[
\left(\frac{s}{2}\right)^2 = 49
\]
4. **Solving for \( s \)**:
Taking the square root:
\[
\frac{s}{2} = 7
\]
Therefore:
\[
s = 2 \times 7 = 14 \text{ in}
\]
Thus, the side length of the base of the square pyramid is **14 inches**.