Question Video: The Side Lengths of 30-60-90 Triangles Mathematics

In triangle 𝐴𝐡𝐢, the length of 𝐡𝐢 is π‘₯. Find, in terms of π‘₯, the length of 𝐴𝐢. Find, in terms of π‘₯, the length of 𝐴𝐡.

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Video Transcript

In triangle 𝐴𝐡𝐢, the length of line 𝐡𝐢 is π‘₯. Find, in terms of π‘₯, the length of line 𝐴𝐢. Find, in terms of π‘₯, the length of line 𝐴𝐡.

When we look at triangle 𝐴𝐡𝐢, we should notice something. First, that angle 𝐴𝐡𝐢 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 𝐡𝐢 has a measure of π‘₯. 𝐡𝐢 is the side opposite the 30-degree angle. So if we consider the ratio one to the square root of three to two, the π‘₯ would go beneath the one. That’s line 𝐡𝐢. Line 𝐴𝐢 is the hypotenuse and line 𝐴𝐡 is the side opposite the 60-degree angle. We’re going to leave the side lengths in terms of π‘₯. 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 π‘₯, then double that will be two π‘₯. And so we can say that the length of line 𝐴𝐢 in terms of π‘₯ is two π‘₯. 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 π‘₯. The length of line 𝐴𝐡 is the square root of three π‘₯. If we add that to the diagram, we have a hypotenuse of two π‘₯ and 𝐴𝐡 equals the square root of three π‘₯. This is because π‘₯, the square root of three π‘₯, and two π‘₯ are in the ratio one to square root of three to two.

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