To find the height of an equilateral triangle with sides measuring [tex]\( 16 \sqrt{3} \)[/tex] units, we can use a geometric relationship specific to equilateral triangles.
An equilateral triangle has all three sides of equal length, and all internal angles are [tex]\( 60^\circ \)[/tex]. One useful property of an equilateral triangle is that the height (or altitude) can divide it into two 30-60-90 right triangles.
For a 30-60-90 triangle, the ratio of the sides opposite these angles is well-known:
- The side opposite the 30° angle is half the hypotenuse.
- The side opposite the 60° angle (which corresponds to the height we need) is [tex]\(\sqrt{3} / 2\)[/tex] times the hypotenuse.
Given that the side length of the equilateral triangle is [tex]\( 16 \sqrt{3} \)[/tex] units, this entire side acts as the hypotenuse of the 30-60-90 triangle.
To find the height [tex]\( h \)[/tex], calculate:
[tex]\[ h = \left(\frac{\sqrt{3}}{2}\right) \times (16 \sqrt{3}) \][/tex]
Let's break it down:
1. Multiply the constants:
[tex]\[ \frac{\sqrt{3}}{2} \times 16 \sqrt{3} \][/tex]
2. Simplify within the multiplication:
[tex]\[ = \frac{\sqrt{3} \cdot 16 \cdot \sqrt{3}}{2} \][/tex]
3. Note that [tex]\(\sqrt{3} \cdot \sqrt{3} = 3\)[/tex]:
[tex]\[ = \frac{16 \cdot 3}{2} \][/tex]
4. Simplify the fraction:
[tex]\[ = \frac{48}{2} \][/tex]
5. Final calculation:
[tex]\[ = 24 \][/tex]
Thus, the height of the equilateral triangle MNO is [tex]\( \boxed{24} \)[/tex] units.
