What is the length of side BC in Figure ?

Options

A. `7.3`
B. `7.7`
C. `8.1`
D. `8.5`

Answer

D

Explanation

You might not think that this is a trigonometry question at first. There's no ''sin,'' ''cos,'', or any other explicit tring function mentioned in the stem. And the triangle's not a right triangle. But trig is the tool you have to use answer this question. This is a case of ''solving a triangle'', that is, finding the length of one or more sides.
With the Law of Sines and the Law of Cosines and a calculator, you can solve almost any triangle. If you know any two angles - which of course means that you know all three - and one side, you can use the Law of Since to find the other two sides. If you know two sides and the angle between them - remember, it must be the angle between them - you can use the Law of Cosines to find the third side. In Example 6, what you're given is two angles and a side, so you'll use the Law of Sines, which says simply that the sines are proportional to the opposite sides. Here the side you're looking for BC, is opposite the `108^(@)`. The side you're given, AC, is opposite the unlabeled angle, which measures `180-108-38=34` degrees. Now you can set up the proportion.
`(BC)/(sin A)=(AC)/(sin B)`
`(BC)/(sin 108^(@))=(5.0)/(sin 34^(@))`
`BC = (5 sin 108^(@))/(sin 34^(@))~~ 8.5`
The answer is (D).