Question

construct an equilateral triangle with each side having length 2k. answers through (a) to (d). (a) what is the measure of each angle? 60° (type a whole number.) (b) label one angle a. drop a perpendicular from a to the side opposite a. two 30° angles are formed at a, and two right triangles are formed. what is the length of the sides opposite the 30° angles? k (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) (c) what is the length of the perpendicular in part (b)? (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall Pythagorean theorem

In a right - triangle formed from the equilateral triangle, the hypotenuse is $2k$ and one side is $k$. Let the length of the perpendicular be $h$. According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 2k$ and $a = k$.

Step2: Solve for $h$

We have $h=\sqrt{(2k)^{2}-k^{2}}=\sqrt{4k^{2}-k^{2}}=\sqrt{3k^{2}}=\sqrt{3}k$.

Answer:

$\sqrt{3}k$