Question

the base of a solid oblique pyramid is an equilateral triangle with a base edge length of 18 inches.

what is the height of the triangular base of the pyramid?

\\(9\sqrt{2}\text{ in.}\\)
\\(9\sqrt{3}\text{ in.}\\)
\\(18\sqrt{2}\text{ in.}\\)
\\(18\sqrt{3}\text{ in.}\\)

Answer:

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<reasoning>

Identify the properties of the base

The problem states that the base of the solid oblique pyramid is an equilateral triangle.
The side length (base edge length) of this equilateral triangle is given as:
\[s = 18\text{ in.}\]

Relate the height to the side length

An equilateral triangle can be split into two congruent \(30^\circ\)-\(60^\circ\)-\(90^\circ\) right triangles by drawing an altitude (height) from one vertex to the opposite side.
In a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) right triangle:

• The hypotenuse is the side length of the equilateral triangle: \(s = 18\text{ in.}\)
• The shorter leg is half of the side length: \(\frac{s}{2} = 9\text{ in.}\)
• The longer leg is the height \(h\) of the triangle.

Calculate the height of the base

Using the ratios of a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) right triangle, the height \(h\) is:
\[h = \frac{s}{2}\sqrt{3}\]

Substitute the given side length \(s = 18\text{ in.}\):
\[h = 9\sqrt{3}\text{ in.}\]
</reasoning>

<answer>
<mcq-option>(A) \(9\sqrt{2}\text{ in.}\)</mcq-option>
<mcq-correct>(B) \(9\sqrt{3}\text{ in.}\)</mcq-correct>
<mcq-option>(C) \(18\sqrt{2}\text{ in.}\)</mcq-option>
<mcq-option>(D) \(18\sqrt{3}\text{ in.}\)</mcq-option>
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