To determine the height of an equilateral triangle given that each side measures \( 34\sqrt{3} \) units, we can follow these steps:
1. Identify the formula for the height of an equilateral triangle:
The height \( h \) of an equilateral triangle with side length \( a \) is given by the formula:
[tex]\[
h = \frac{\sqrt{3}}{2} \times a
\][/tex]
2. Substitute the given side length into the formula:
Here, we are given that \( a = 34\sqrt{3} \) units.
So, substituting this into the formula, we have:
[tex]\[
h = \frac{\sqrt{3}}{2} \times (34\sqrt{3})
\][/tex]
3. Simplify the expression:
Let's simplify the expression step-by-step:
[tex]\[
h = \frac{\sqrt{3}}{2} \times (34\sqrt{3})
\][/tex]
[tex]\[
h = \left(\frac{\sqrt{3}}{2}\right) \times 34 \times \sqrt{3}
\][/tex]
[tex]\[
h = 34 \times \left(\frac{\sqrt{3} \times \sqrt{3}}{2}\right)
\][/tex]
[tex]\[
h = 34 \times \left(\frac{3}{2}\right)
\][/tex]
[tex]\[
h = 34 \times 1.5
\][/tex]
[tex]\[
h = 51
\][/tex]
Thus, the height of the equilateral triangle is \( 51 \) units.
So, out of the given multiple-choice options, the correct answer is:
[tex]\[
\boxed{51 \text{ units}}
\][/tex]
