Video: Using the Sine Ratio to Find the Length of the Hypotenuse

Video Transcript

Find the length of line segment 𝐴𝐶 given 𝐴𝐵𝐶 is a right-angled 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. First, we have a right triangle. The right angle is at 𝐵. So we label our right angle 𝐵. And then we add 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 talking about is angle 𝐶. And if we think of our trig ratio acronym SOHCAHTOA, we know that the sin of an angle is equal to the opposite over the hypotenuse. And so if the angle we’re considering is 𝐶, the opposite side length will be the side 𝐴𝐵. And the hypotenuse is always the side opposite the right angle. And so that relationship is 9 to 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 as 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 could say that the hypotenuse must measure 32 centimeters. Line segment 𝐴𝐶 is the hypotenuse. And that measures 32 centimeters.