𝐴𝐵𝐶 is a triangle where 𝑏 is equal to 28 centimeters, 𝑐 is equal to 17 centimeters, and the measure of angle 𝐶 is 32 degrees. Find all the possible values for the perimeter giving the answer to two decimal places.
We begin by drawing a quick sketch of what the triangle may look like. We are told that 𝑏 is equal to 28 centimeters, 𝑐 is equal to 17 centimeters, and the measure of angle 𝐶 is 32 degrees. We are asked to find all the possible values for the perimeter. In order to do this, we firstly need to find the length of the side opposite angle 𝐴, which we have labeled lowercase 𝑎. We can do this using the law of sines. However, we’ll firstly need to calculate the measure of angle 𝐴 and prior to that the measure of angle 𝐵.
Using the alternative version of the law of sines, we have sin 𝐵 over 28 is equal to sin of 32 degrees over 17. Multiplying through by 28, we have sin 𝐵 is equal 0.8728 and so on. We can then take the inverse sine of both sides. 𝐵 is therefore equal to 60.7866 and so on. One possible measure of angle 𝐵 is 60.7866 and so on degrees. Next, we recall that since the sin of 180 degrees minus 𝜃 is equal to sin 𝜃, there is another angle less than 180 degrees that has a sine equal to 0.8728 and so on. This is the supplementary angle for 𝐵. Subtracting 60.7866 and so on from 180 gives us 119.2133 and so on.
We have a second possible measure of angle 𝐵 equal to 119.2133 and so on degrees and two possible triangles as shown. Since angles in a triangle sum to 180 degrees, we are now in a position to calculate the possible measures of angle 𝐴. Firstly, we find the sum of 32 and 60.7866 and so on and then subtract this from 180. This gives us 𝐴 is equal to 87.2133 and so on. This is the first possible measure of angle 𝐴 in degrees. Repeating this for our second triangle, we have 𝐴 is equal to 28.7866 and so on. Once again, this is a possible measure of angle 𝐴 in degrees.
We are now in a position to calculate the possible values of side length 𝑎. We do this using the law of sines once again. We have 𝑎 over sin of 87.2133 and so on degrees is equal to 17 over sin of 32 degrees. Rearranging this equation as shown, we have 𝑎 is equal to 32.04 centimeters to two decimal places. Once again, we will repeat this for our second scenario. This time, we have 𝑎 is equal to 15.45 centimeters to two decimal places. Now that we have the two possible values of side length 𝑎, we are in a position to calculate the possible values of the perimeter.
Firstly, we need to find the sum of 28, 17, and 32.04. This is equal to 77.04, so one possible value of the perimeter is 77.04 centimeters. Secondly, we need to find the sum of 28, 17, and 15.45. This is equal to 60.45, and the second possible value of the perimeter is 60.45 centimeters. If triangle 𝐴𝐵𝐶 has measurements 𝑏 is equal to 28 centimeters, 𝑐 is equal to 17 centimeters, and the measure of angle 𝐶 is 32 degrees, the two possible perimeters of the triangle are 77.04 centimeters and 60.45 centimeters.