Video Transcript
A farmer wants to fence off a triangular piece of land. The lengths of two sides of the fence are 72 meters and 55 meters, and the angle between them is 83 degrees. Find the perimeter of the fence giving your answer to the nearest meter.
We haven’t been given a diagram in this question, so we should begin by sketching the triangular piece of land using the information given. This field has sides of lengths 72 meters and 55 meters. And the angle between them, which we might refer to as the enclosed or included angle, is 83 degrees. And then we’ll complete the triangle. We’re asked to find the perimeter of the fence, which means we need to calculate the length of the third side of the triangle. This is a non-right triangle in which we’ve been given the lengths of two sides and the measure of their included angle. This is the right combination of information for us to be able to apply the law of cosines to calculate the length of the third side. Let’s recall its definition.
The law of cosines states that in a non-right triangle with sides labeled using the lowercase letters 𝑎, 𝑏, and 𝑐 and corresponding opposite angles labeled using the uppercase letters 𝐴, 𝐵, and 𝐶, 𝑎 squared is equal to 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 cos 𝐴. This means that if we know the lengths of two sides and the measure of their included angle, as we do here, then we can calculate the length of the third side by applying the law of cosines.
We may find it helpful to label the sides and angles of our triangle using the letters corresponding to those used in the law of cosines. However, this isn’t essential if we’re familiar with the structure of the law of cosines. If we can remember that 𝑏 and 𝑐 represent the two known side lengths and 𝐴 represents their included angle, then we can substitute the given values directly into the law of cosines without explicitly needing to label the sides and angles using letters.
Either way, substituting 55 and 72 for the side lengths and 83 degrees for the angle between them, we have 𝑎 squared equals 55 squared plus 72 squared minus two times 55 times 72 times cos of 83 degrees. We can evaluate some parts of this to give 𝑎 squared equals 3025 plus 5184 minus 7920 cos of 83 degrees. And evaluating further, making sure our calculators are in degree mode, gives 𝑎 squared is equal to 7243.794. Now remember, this is the value of 𝑎 squared, not the value of 𝑎. So we need to solve for 𝑎 by square rooting. We can ignore the negative solution because 𝑎 represents a length, so we have 𝑎 is equal to 85.110.
So we found the length of the third side of this triangle by applying the law of cosines. And finally, we need to find its perimeter. Adding the three side lengths together gives 72 plus 55 plus 85.110, which is equal to 212.110. The question specifies that we should give our answer to the nearest meter. So we round this value, and we find that the perimeter of the fence to the nearest meter is 212 meters.
