Question

Consider an equilateral triangle with each side having length 2k.

(a) What is the measure of each angle?
(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?
(c) What is the length of the perpendicular in part (b)?
(d) From the results of parts (a)-(c), complete the following statement: In a 30⁰ - 60⁰ 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.

Answer 1

In an equilateral triangle with side lengths of 2k, each angle measures 60 degrees. Dropping a perpendicular creates two 30-60-90 right triangles with the sides opposite the 30-degree angles measuring k, and the perpendicular measuring k√3. The hypotenuse is twice the shorter leg, and the longer leg is √3 times the shorter leg.

Step-by-step explanation:

Answering the question about an equilateral triangle with side lengths of 2k:

1. Each angle of an equilateral triangle measures 60 degrees, since all angles in any triangle add up to 180 degrees and all angles are equal in an equilateral triangle.
2. When you drop a perpendicular from angle A to the opposite side, you create two 30-60-90 right triangles. The side opposite the 30-degree angles, which is also half of the triangle's base, will be k units long.
3. The length of the perpendicular can be found using the Pythagorean theorem (height2 + k2 = (2k)2). Solving for the height, we find it to be k√3.
4. From the results, the hypotenuse is always twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. The shorter leg is opposite the 30-degree angle, and the longer leg is opposite the 60-degree angle.