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
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.