𝐴𝐵𝐶 is a triangle, where 𝑎 is equal to 14 centimeters, 𝑏 is equal to nine centimeters, and the measure of angle 𝐶 is 70 degrees. Find the perimeter of the triangle, giving the answer to the nearest centimeter.
We will begin by sketching triangle 𝐴𝐵𝐶. When labeling our triangle, the lowercase letters represent the lengths of the sides and the capital letters represent the measure of the angles. We are told that side length 𝑎 is equal to 14 centimeters. Side length 𝑏 is equal to nine centimeters. The measure of angle 𝐶 is equal to 70 degrees. We are asked to calculate the perimeter. And we know this is the distance around the outside of the triangle. As we know two of the side lengths, we need to calculate the length of the third side, lowercase 𝑐. Once we have done that, we can find the sum of all three sides.
We can calculate the length of the missing side using the law of cosines. This states that 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared minus two 𝑎𝑏 multiplied by the cos of angle 𝑐. Substituting in our values without their units, we have 𝑐 squared is equal to 14 squared plus nine squared minus two multiplied by 14 multiplied by nine multiplied by the cos of 70 degrees. Typing the right-hand side into our calculator gives us 190.8109 and so on. As this is equal to 𝑐 squared, we can square root both sides to calculate the value of 𝑐. 𝑐 is equal to 13.8134 and so on.
We are asked to give our answer to the nearest centimeter. And we would usually do this rounding at the end of the question. However, in this case, as the other two lengths are integers, we can round now. As the first decimal place is five or greater, we round up. To the nearest centimeter, 𝑐 is equal to 14 centimeters. We can now calculate the perimeter by adding the lengths of the three sides. We need to find the sum of 14, nine, and 14. This is equal to 37. The perimeter of the triangle to the nearest centimeter is 37 centimeters.