Question Video: The Side Lengths of 30-60-90 Triangles Mathematics • 11th Grade

Video Transcript

In triangle 𝐴𝐵𝐶, the length of line 𝐵𝐶 is 𝑥. Find, in terms of 𝑥, the length of line 𝐴𝐶. Find, in terms of 𝑥, the length of line 𝐴𝐵.

When we look at triangle 𝐴𝐵𝐶, we should notice something. First, that angle 𝐴𝐵𝐶 is a right angle and then that the other two angles are 60 degrees and 30 degrees. This is a triangle we call a 30-60-90 triangle. And we know that this triangle has a ratio of side lengths one to the square root of three to two, where the one side is the side opposite the 30-degree angle. The square root of three side is the side opposite the 60-degree angle. And the two side is the hypotenuse.

We were told that line 𝐵𝐶 has a measure of 𝑥. 𝐵𝐶 is the side opposite the 30-degree angle. So if we consider the ratio one to the square root of three to two, the 𝑥 would go beneath the one. That’s line 𝐵𝐶. Line 𝐴𝐶 is the hypotenuse and line 𝐴𝐵 is the side opposite the 60-degree angle. We’re going to leave the side lengths in terms of 𝑥. And so we need to think about how this ratio works.

How do we go from one to two? We multiply it by two. This means the hypotenuse is twice as long as the shortest side in a 30-60-90 triangle. If the shortest side in our triangle measures 𝑥, then double that will be two 𝑥. And so we can say that the length of line 𝐴𝐶 in terms of 𝑥 is two 𝑥. To find the remaining side length, we’ll ask a similar question. How do we go from one to the square root of three?

Well, you multiply by the square root of three. The side length opposite the 60-degree angle is the square root of three times more than the shortest side, which means it’s the square root of three times 𝑥. The length of line 𝐴𝐵 is the square root of three 𝑥. If we add that to the diagram, we have a hypotenuse of two 𝑥 and 𝐴𝐵 equals the square root of three 𝑥. This is because 𝑥, the square root of three 𝑥, and two 𝑥 are in the ratio one to square root of three to two.