An equilateral triangle has sides measuring [tex]$34 \sqrt{3}$[/tex] units. What is the height of the triangle?

A. 17 units
B. 34 units
C. 51 units
D. 68 units



Answer :

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]