# Question Video: Using the Sine Ratio to Find the Length of the Hypotenuse Mathematics • 11th Grade

Find the length of the line segment ๐ด๐ถ, given that ๐ด๐ต๐ถ is a right triangle at ๐ต, where sin ๐ถ = 9/16 and ๐ด๐ต = 18 cm.

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### Video Transcript

Find the length of the line segment ๐ด๐ถ, given that ๐ด๐ต๐ถ is a right triangle at ๐ต, where sin of ๐ถ equals nine over 16 and ๐ด๐ต equals 18 centimeters.

In this case, the first step should be to sketch a right triangle that meets these conditions. We have a right triangle. The right angle is at ๐ต, so we label the right angle ๐ต. And then we add on ๐ด and ๐ถ. Weโre told that ๐ด๐ต measures 18 centimeters. And then we have this other piece of information that sin of ๐ถ equals nine over 16. This tells us that the angle weโre working with is angle ๐ถ. And if we think of our acronym SOH CAH TOA, we know that the sine of an angle is equal to the opposite over the hypotenuse.

If the angle weโre considering is ๐ถ, the opposite will be the side ๐ด๐ต, and the hypotenuse is always the side opposite the right angle. Itโs the side ๐ด๐ถ. And so that ratio is nine over 16. The key thing to remember here is that these relationships are ratios. And so sin of angle ๐ถ tells us that for every nine units on the opposite side length, there will be 16 units on the hypotenuse side length.

So we can say that if there are 18 centimeters on the opposite side, we know that nine times two equals 18. And when dealing with ratios or fractions, if we multiply by two in the numerator, we need to multiply by two in the denominator. 16 times two is 32. And so we can say that if the opposite is 18, the hypotenuse must be 32. Line segment ๐ด๐ถ is the hypotenuse, and it measures 32 centimeters.