𝐴𝐵𝐶 is a triangle, where the measure of angle 𝐴 is 46 degrees, 11 minutes, and 17 seconds; the measure of angle 𝐵 is 27 degrees, four minutes, 46 seconds; and length 𝐴 is 21.4 centimeters. Find the length of the shortest side of 𝐴𝐵𝐶, giving the answer to one decimal place.
Let’s begin by sketching triangle 𝐴𝐵𝐶. We are told that the measure of angle 𝐴 is 46 degrees, 11 minutes, and 17 seconds and the measure of angle 𝐵 is 27 degrees, four minutes, 46 seconds. We are also told that the side length 𝐴 is equal to 21.4 centimeters. This will be the side length 𝐵𝐶 that is opposite angle 𝐴.
We are asked to calculate the length of the shortest side of the triangle. We know that the shortest side of a triangle is opposite the smallest angle. Since angles in a triangle sum to 180 degrees, we know that angle 𝐶 will be greater than angle 𝐴 and angle 𝐵, which means that angle 𝐵 is the smallest angle in the triangle.
Since we are given the measures of two angles in the triangle, together with the length of one of the sides, and we need to calculate the length of a further side, we can use the sine rule or law of sines. This states that 𝐴 over sin 𝐴 is equal to 𝐵 over sin 𝐵, which is equal to 𝐶 over sin 𝐶. Substituting in the measures of angle 𝐴 and 𝐵 together with side length 𝐴, we have 𝐵 over sin of 27 degrees, four minutes, and 46 seconds is equal to 21.4 over sin of 46 degrees, 11 minutes, and 17 seconds. Multiplying through by sin 𝐵, we have the following equation.
Using the degrees, minutes, and seconds button, we can type the right-hand side directly into our calculator, giving us 𝐵 is equal to 13.500 and so on, which to one decimal place is 13.5. The length of the shortest side of triangle 𝐴𝐵𝐶 is 13.5 centimeters.