Question

a certain triangle has a 30° angle and a 60° angle. which must be a true statement about the triangle?

a. the longest side is √3 times as long as the shortest side.
b. the second - longest side is twice as long as the shortest side.
c. the longest side is twice as long as the shortest side.
d. two sides of the triangle have the same length.

Explanation:

Step1: Find the third - angle

The sum of angles in a triangle is 180°. Given two angles 30° and 60°, the third angle is 180°-(30° + 60°)=90°. So it's a 30 - 60 - 90 triangle.

Step2: Recall side - length relationships

In a 30 - 60 - 90 triangle, if the side opposite the 30° angle is \(a\), the side opposite the 60° angle is \(a\sqrt{3}\), and the side opposite the 90° angle (the hypotenuse, the longest side) is \(2a\). The shortest side is opposite the 30° angle and the longest side (hypotenuse) is twice as long as the shortest side.

Answer:

C. The longest side is twice as long as the shortest side.