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Problem
Each vertical cross-section of the triangular prism shown below is an isosceles triangle.
A triangular prism. The triangular face has a base of four units and a height of five units. One of the non-base sides of the triangular face is labeled S.



What is the slant height,

, of the triangular prism?
Round your answer to the nearest tenth.
The slant height is
units.

Answer 1

To find the slant height \( S \) of the isosceles triangle base of the triangular prism, we can use the given dimensions of the triangle. The base of the triangle is 4 units, and the height is 5 units.

1. **Identify the necessary components:**
- The base \( b = 4 \) units.
- The height \( h = 5 \) units.

2. **Calculate half the base:**
\[
\text{Half of the base} = \frac{b}{2} = \frac{4}{2} = 2 \text{ units}
\]

3. **Use the Pythagorean theorem to find the slant height (side length \( S \)):**
The slant height forms a right triangle along with half the base and the height. According to the Pythagorean theorem:
\[
S^2 = \left(\text{Half of the base}\right)^2 + \left(\text{Height}\right)^2
\]
Substituting the known values:
\[
S^2 = (2)^2 + (5)^2 = 4 + 25 = 29
\]

4. **Calculate \( S \):**
\[
S = \sqrt{29}
\]

5. **Calculate the numerical value and round to the nearest tenth:**
\[
S \approx 5.385 \implies S \approx 5.4 \text{ (rounded to the nearest tenth)}
\]

Therefore, the slant height \( S \) of the triangular prism is approximately **5.4 units**.