Question

find the height of the equilateral triangle. (you can use any method. if using pythagorean theorem, remember su is being cut in half. if using trig, remember that all angles in an equilateral triangle are 60°but ∠t is cut in half.) leave your answer in simplest radical form.
options: 8√2, 3√4, 8√10, 4√3

Explanation:

Step1: Identify the right triangle

The height (let's call it \( h \)) of the equilateral triangle splits it into two right triangles. The hypotenuse of each right triangle is 8 (side of the equilateral triangle), and the base of each right triangle is \( \frac{8}{2} = 4 \) (since the height bisects the base \( SU \)).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right triangle with hypotenuse \( c \), and legs \( a \) and \( b \), \( a^2 + b^2 = c^2 \). Here, \( c = 8 \), \( a = 4 \), and \( b = h \) (the height we want to find). So we have:

$$ 4^2 + h^2 = 8^2 $$

Step3: Solve for \( h \)

First, calculate \( 4^2 = 16 \) and \( 8^2 = 64 \). Then:

$$ 16 + h^2 = 64 $$

Subtract 16 from both sides:

$$ h^2 = 64 - 16 = 48 $$

Take the square root of both sides:

$$ h = \sqrt{48} $$

Simplify \( \sqrt{48} \): \( \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \)

Answer:

\( 4\sqrt{3} \) (corresponding to the option with \( 4\sqrt{3} \))