Question

triangle mno is an equilateral triangle with sides measuring $16\sqrt{3}$ units.
what is the height of the triangle?
12 units
24 units
36 units
72 units

Explanation:

Step1: Recall properties of equilateral triangle

In an equilateral triangle, the height (altitude) splits the triangle into two congruent 30 - 60 - 90 right triangles. The side of the equilateral triangle is the hypotenuse of the right triangle, and half of the base of the equilateral triangle is the shorter leg of the 30 - 60 - 90 triangle.
The length of the base of the equilateral triangle is \(a = 16\sqrt{3}\), so half of the base \(MR=\frac{a}{2}=\frac{16\sqrt{3}}{2} = 8\sqrt{3}\).

Step2: Use Pythagorean theorem or 30 - 60 - 90 triangle ratios

For a 30 - 60 - 90 triangle, the ratio of the sides is \(1:\sqrt{3}:2\) (shorter leg : longer leg : hypotenuse). Here, the hypotenuse (side of equilateral triangle) is \(16\sqrt{3}\), and the shorter leg is \(8\sqrt{3}\). Let the height (longer leg) be \(h\).
Using the Pythagorean theorem \(h=\sqrt{(16\sqrt{3})^{2}-(8\sqrt{3})^{2}}\)
First, calculate \((16\sqrt{3})^{2}=16^{2}\times(\sqrt{3})^{2}=256\times3 = 768\)
and \((8\sqrt{3})^{2}=8^{2}\times(\sqrt{3})^{2}=64\times3 = 192\)
Then \(h=\sqrt{768 - 192}=\sqrt{576}=24\)
Or using the 30 - 60 - 90 ratio: the longer leg (height) is \(\sqrt{3}\) times the shorter leg. So \(h = 8\sqrt{3}\times\sqrt{3}=8\times3 = 24\)

Answer:

24 units