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? (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall properties of equilateral - triangle

In an equilateral triangle with side - length \(s = 2k\), when a perpendicular is dropped from a vertex to the opposite side, it bisects the opposite side.

Step2: Identify the right - triangle formed

The right - triangle has a hypotenuse equal to the side of the equilateral triangle (\(s = 2k\)) and the angle opposite the side we want to find is \(30^{\circ}\).

Step3: Use the 30 - 60 - 90 triangle ratio

In a 30 - 60 - 90 triangle, if the hypotenuse is \(c\) and the side opposite the \(30^{\circ}\) angle is \(a\), then \(a=\frac{1}{2}c\). Here, \(c = 2k\), so \(a = k\).

Answer:

\(k\)