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
(image of equilateral triangle mno with height nr, base mo labeled $16\sqrt{3}$, sides mn and no labeled $16\sqrt{3}$)

Explanation:

Step1: Split base into two equal parts

The base $MO = 16\sqrt{3}$, so $MR = \frac{16\sqrt{3}}{2} = 8\sqrt{3}$ units.

Step2: Apply Pythagorean theorem

Let height $NR = h$. For right triangle $MRN$:
$$h^2 + (8\sqrt{3})^2 = (16\sqrt{3})^2$$

Step3: Calculate squared terms

$$h^2 + 8^2 \times 3 = 16^2 \times 3$$
$$h^2 + 64 \times 3 = 256 \times 3$$
$$h^2 + 192 = 768$$

Step4: Solve for $h^2$

$$h^2 = 768 - 192 = 576$$

Step5: Find square root of $h^2$

$$h = \sqrt{576} = 24$$

Answer:

24 units