To find the height of the triangular base of a pyramid when the base is an equilateral triangle with an edge length of [tex]\( s \)[/tex] units, we need to use the geometry of an equilateral triangle.
An equilateral triangle has three sides of equal length and all internal angles equal to [tex]\( 60^\circ \)[/tex]. The height [tex]\( h \)[/tex] of an equilateral triangle can be derived using basic trigonometry or geometric properties.
For an equilateral triangle, the height divides the triangle into two 30-60-90 right triangles. The relationship between the sides of a 30-60-90 triangle is well-established: the length of the side opposite the [tex]\( 30^\circ \)[/tex] angle is half the hypotenuse, and the length of the side opposite the [tex]\( 60^\circ \)[/tex] angle is [tex]\( \frac{\sqrt{3}}{2} \)[/tex] times the hypotenuse.
For our equilateral triangle with side length [tex]\( s \)[/tex]:
- The base of each 30-60-90 triangle (half the side length of the equilateral triangle) is [tex]\( \frac{s}{2} \)[/tex].
- The height [tex]\( h \)[/tex], opposite the [tex]\( 60^\circ \)[/tex] angle, is:
[tex]\[
h = \frac{\sqrt{3}}{2} \cdot s
\][/tex]
Therefore, the height of the triangular base of the pyramid is:
[tex]\[
h = \frac{s \sqrt{3}}{2}
\][/tex]
Given the possible choices:
[tex]\[
\frac{5}{2} \sqrt{2}, \quad \frac{5}{2} \sqrt{3}, \quad s \sqrt{2}, \quad 5 \sqrt{3}
\][/tex]
The correct expression matches [tex]\(\frac{s \sqrt{3}}{2}\)[/tex]. To check against the numerical results:
1. [tex]\(\frac{5}{2} \sqrt{2} \approx 3.5355339059327378\)[/tex]
2. [tex]\(\frac{5}{2} \sqrt{3} \approx 4.330127018922193\)[/tex]
3. [tex]\(s \sqrt{2} \approx 1.4142135623730951\)[/tex] (assuming [tex]\(s = 1\)[/tex])
4. [tex]\(5 \sqrt{3} \approx 8.660254037844386\)[/tex]
5. [tex]\(\frac{s \sqrt{3}}{2} \approx 0.8660254037844386\)[/tex] (assuming [tex]\(s = 1\)[/tex])
The correct height of the triangular base, as derived, is approximately [tex]\( 0.8660254037844386 \)[/tex], which corresponds to [tex]\(\frac{s \sqrt{3}}{2}\)[/tex].
Thus, none of the provided choices directly represent [tex]\(\frac{s \sqrt{3}}{2}\)[/tex] as the height.
This suggests that the correct representation of the height of the triangular base of the pyramid is [tex]\(\frac{s \sqrt{3}}{2}\)[/tex], however, it is noticeable that this representation was not presented among the discussed options explicitly; hence, among the given choices, selection none makes more sense logically unless given the correct equation correctly.
