Question Video: Using the Sine Rule to Calculate the Perimeter of a Triangle Mathematics

Video Transcript

𝐴𝐵𝐶 is a triangle where the measure of angle 𝐴 is 59 degrees, 55 minutes, and 16 seconds; the measure of angle 𝐵 is 54 degrees, 41 minutes, and 27 seconds; and 𝑐 is equal to 16.6 centimeters. Find the perimeter of 𝐴𝐵𝐶, giving the answer to one decimal place.

We begin by sketching triangle 𝐴𝐵𝐶 as shown. We are told that the measure of angle 𝐴 is 59 degrees, 55 minutes, and 16 seconds. The measure of angle 𝐵 is 54 degrees, 41 minutes, and 27 seconds. We know that side lengths lowercase 𝑎, 𝑏, and 𝑐 are opposite their corresponding angles. This means that the side with length 16.6 centimeters is opposite angle 𝐶. We are asked to calculate the perimeter of the triangle, and we know this will be the distance around the outside. It is the sum of side lengths 𝑎, 𝑏, and 𝑐.

We can calculate the lengths of 𝑎 and 𝑏 using the sine rule, which states that 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵, which is equal to 𝑐 over sin 𝐶. Substituting in the values we are given, we have 𝑎 over sin of 59 degrees, 55 minutes, and 16 seconds is equal to 𝑏 over sin of 54 degrees, 41 minutes, and 27 seconds is equal to 16.6 over sin 𝐶. We know that angles in a triangle sum to 180 degrees. This means that the measure of angle 𝐶 is equal to 180 degrees minus the sum of the measure of angles 𝐴 and 𝐵. We need to subtract 114 degrees, 36 minutes, and 43 seconds from 180 degrees. This means that angle 𝐶 is equal to 65 degrees, 23 minutes, and 17 seconds.

We now know the measure of all three angles in the triangle. And we’re in a position to use the sine rule to calculate lengths 𝑎 and 𝑏. Since 𝑎 over sin 59 degrees, 55 minutes, and 16 seconds is equal to 16.6 over sin 65 degrees, 23 minutes, and 17 seconds, then 𝑎 is equal to the following. Using the degrees, minutes, and seconds button, we can type the right-hand side into our calculator, giving us 15.800 and so on. To one decimal place, 𝑎 is equal to 15.8 centimeters.

We can repeat this process to find the length of side 𝑏. This is equal to 16.6 divided by sin of 65 degrees, 23 minutes, 17 seconds multiplied by the sin of 54 degrees, 41 minutes, and 27 seconds. This is equal to 14.900 and so on, which rounds to 14.9 centimeters. We can now calculate the perimeter of the triangle by finding the sum of 15.8, 14.9, and 16.6. This is equal to 47.3. The perimeter of triangle 𝐴𝐵𝐶 is 47.3 centimeters.