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

In triangle ๐ด๐ต๐ถ, the length of ๐ต๐ถ is ๐‘ฅ. Find, in terms of ๐‘ฅ, the length of ๐ด๐ถ. Find, in terms of ๐‘ฅ, the length of ๐ด๐ต.

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Video Transcript

In triangle ๐ด๐ต๐ถ, the length of line ๐ต๐ถ is ๐‘ฅ. Find, in terms of ๐‘ฅ, the length of line ๐ด๐ถ. Find, in terms of ๐‘ฅ, the length of line ๐ด๐ต.

When we look at triangle ๐ด๐ต๐ถ, we should notice something. First, that angle ๐ด๐ต๐ถ 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 ๐ต๐ถ has a measure of ๐‘ฅ. ๐ต๐ถ is the side opposite the 30-degree angle. So if we consider the ratio one to the square root of three to two, the ๐‘ฅ would go beneath the one. Thatโ€™s line ๐ต๐ถ. Line ๐ด๐ถ is the hypotenuse and line ๐ด๐ต is the side opposite the 60-degree angle. Weโ€™re going to leave the side lengths in terms of ๐‘ฅ. 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 ๐‘ฅ, then double that will be two ๐‘ฅ. And so we can say that the length of line ๐ด๐ถ in terms of ๐‘ฅ is two ๐‘ฅ. To find the remaining side length, weโ€™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 itโ€™s the square root of three times ๐‘ฅ. The length of line ๐ด๐ต is the square root of three ๐‘ฅ. If we add that to the diagram, we have a hypotenuse of two ๐‘ฅ and ๐ด๐ต equals the square root of three ๐‘ฅ. This is because ๐‘ฅ, the square root of three ๐‘ฅ, and two ๐‘ฅ are in the ratio one to square root of three to two.

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