Question

the base of a solid oblique pyramid is an equilateral triangle with an edge length of s units.

which expression represents the height of the triangular base of the pyramid?

\\(\frac{s}{2}\sqrt{2}\\) units
\\(\frac{s}{2}\sqrt{3}\\) units
\\(s\sqrt{2}\\) units
\\(s\sqrt{3}\\) units

Answer:

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<reasoning>

Identify the properties of the base

Using the Equilateral Triangle Properties knowledge point

• The base of the oblique pyramid is an equilateral triangle.
• The side length (edge length) of this equilateral triangle is \(s\).
• All three interior angles of an equilateral triangle are \(60^\circ\).

Apply properties of special right triangles

Using the Special Right Triangles knowledge point

• Drawing an altitude (height, \(h\)) from one vertex of the equilateral triangle to the opposite side splits the triangle into two congruent \(30^\circ\)-\(60^\circ\)-\(90^\circ\) right triangles.
• The hypotenuse of each right triangle is the side length \(s\).
• The shorter leg is half of the base: \(\frac{s}{2}\).
• The longer leg is the height \(h\), which is \(\sqrt{3}\) times the shorter leg:

\[
h = \frac{s}{2}\sqrt{3}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{s}{2}\sqrt{2}\) units</mcq-option>
<mcq-correct>(B) \(\frac{s}{2}\sqrt{3}\) units</mcq-correct>
<mcq-option>(C) \(s\sqrt{2}\) units</mcq-option>
<mcq-option>(D) \(s\sqrt{3}\) units</mcq-option>
</answer>

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