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

Find the length of the line segment 𝐴𝐶, given that 𝐴𝐵𝐶 is a right triangle at 𝐵, where sin 𝐶 = 9/16 and 𝐴𝐵 = 18 cm.

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Video Transcript

Find the length of the line segment 𝐴𝐶, given that 𝐴𝐵𝐶 is a right triangle at 𝐵, where sin of 𝐶 equals nine over 16 and 𝐴𝐵 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 𝐵, so we label the right angle 𝐵. And then we add on 𝐴 and 𝐶. We’re told that 𝐴𝐵 measures 18 centimeters. And then we have this other piece of information that sin of 𝐶 equals nine over 16. This tells us that the angle we’re working with is angle 𝐶. 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 we’re considering is 𝐶, the opposite will be the side 𝐴𝐵, and the hypotenuse is always the side opposite the right angle. It’s the side 𝐴𝐶. 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 𝐶 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 𝐴𝐶 is the hypotenuse, and it measures 32 centimeters.

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