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)? √3k (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) (d) from the results of parts (a) - (c), complete the following statement. in a 30° - 60° right triangle, the hypotenuse is always times as long as the shorter leg, and the longer leg has a length that is times as long as that of the shorter leg. also, the shorter leg is opposite the ° angle, and the longer leg is opposite the ° angle. (simplify your answers, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall properties of 30 - 60 - 90 triangle

In a 30 - 60 - 90 right - triangle formed from the equilateral triangle, if the shorter leg (opposite 30°) has length \(x\), the hypotenuse has length \(2x\) and the longer leg (opposite 60°) has length \(\sqrt{3}x\).

Step2: Analyze hypotenuse - shorter leg ratio

Since the hypotenuse \(h = 2x\) and the shorter leg \(s=x\), the hypotenuse is 2 times as long as the shorter leg.

Step3: Analyze longer leg - shorter leg ratio

Since the longer leg \(l=\sqrt{3}x\) and the shorter leg \(s = x\), the longer leg has a length that is \(\sqrt{3}\) times as long as the shorter leg.

Step4: Identify angles opposite legs

The shorter leg is opposite the 30° angle and the longer leg is opposite the 60° angle.

Answer:

In a 30° - 60° right triangle, the hypotenuse is always 2 times as long as the shorter leg, and the longer leg has a length that is \(\sqrt{3}\) times as long as that of the shorter leg. Also, the shorter leg is opposite the 30° angle, and the longer leg is opposite the 60° angle.