Question Video: Using the Law of Cosines to Calculate the Unknown Length of a Triangle Mathematics

Video Transcript

In the given figure, find 𝑥. Give your answer to two decimal places.

We are given the measure of one of the angles of the triangle and the lengths of two of the sides, and we are required to find the length of the third side which is opposite the given angle. This looks like a job for the cosine rule. The cosine rule relates the lengths of all three sides of a triangle to the measures of one of the angles of the triangle. Choosing that angle to be at the vertex 𝐴, we have that 𝑎 squared equals 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 times the cosine of that measure 𝐴.

So let’s compare this to what we have in our diagram. The length of the side opposite vertex 𝐴 is 𝑥, our unknown. The value of 𝑏, the length of the side opposite vertex 𝐵 is 12. In the same way, we can see that the value of 𝑐 is seven and the measure of the angle at the vertex 𝐴 is 51 degrees. The right-hand side is something that we can put into our calculators.

Doing so, we get a value of 87.274 dot dot dot. And taking square roots on both sides and knowing that 𝑥 must be positive as it is the length of a side, we get that 𝑥 is 9.3420 dot dot dot. And rounding this value to two decimal places as required, we get that 𝑥 is equal to 9.34.