Question Video: Using the Law of Sines to Calculate an Unknown Length given the Perimeter and Ratio between the Angles of a Triangle Mathematics

Video Transcript

𝐴𝐵𝐶 is a triangle with a perimeter of 49 centimeters where the ratio between the measure of the angle at 𝐴, the measure of the angle at 𝐵, and the measure of the angle at 𝐶 is nine to five to four. Find the length of the smallest side, giving the answer to two decimal places.

Since the sum of the interior angles in a triangle is 180 degrees, we can begin by sharing 180 degrees into the ratio nine to five to four. To do this, we first find the total number of parts by adding together each number in the ratio. Nine plus five plus four is 18. Since the measure of the angle at 𝐴 is worth nine of these parts, it’s worth nine 18ths of 180 degrees. Nine 18ths simplifies to one-half. And one-half of 180 degrees is 90. So the measure of the angle at 𝐴 is 90 degrees.

Similarly, the measure of the angle at 𝐵 can be found by finding five 18ths of 180. We can find one 18th by dividing 180 by 18. That’s 10. Five 18ths is five times the size of this. So it’s 50. And the measure of the angle at 𝐵 is 50 degrees. Finally, the measure of the angle at 𝐶 is calculated by finding four 18ths of 180. Since one 18ths was 10 degrees, four 18ths is four times this. That’s 40 degrees. So, we have a right-angled triangle for which we know the measure of all of its angles.

Don’t be fooled into thinking we need to use right angle trigonometry to solve this problem though. We are given that the triangle has a perimeter of 49 centimeters. So that tells us that 𝑎 plus 𝑏 plus 𝑐 must be equal to 49. And, instead, we’re going to use the law of sines to create further expressions in terms of 𝐴, 𝐵, and 𝐶. The law of sines says that 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵, which is equal to 𝑐 over sin 𝐶. And this is sometimes written as sin 𝐴 over 𝑎 equal sin 𝐵 over 𝑏, which equals sin 𝐶 over 𝑐. We can use either of these forms. However, since we’re trying to find missing lengths, we should use the first form. This will reduce the amount of rearranging we need to do.

Let’s substitute what we know into this formula. That gives us 𝑎 over sin 90 equals 𝑏 over sin 50, which equals 𝑐 over sin 40. Let’s split this up a bit. We’re trying to find the shortest side in the triangle, which is the side opposite the smallest angle. The smallest angle is 𝐶. So the side opposite that is denoted by lowercase 𝑐. What we need to do then is form an equation for 𝑎 and 𝑏 in terms of 𝑐. Let’s start with 𝑎. 𝑎 over sin 90 equals 𝑐 over sin 40. We know that sin 90 is equal to one. So, in fact, 𝑎 is equal to 𝑐 over sin 40. We also know that 𝑏 over sin 50 is equal to 𝑐 over sin 40. We can multiply both sides of this equation by sin 50 to get an equation for 𝑏 in terms of 𝑐. That’s 𝑏 is equal to 𝑐 sin 50 over sin 40.

We now have two expressions in terms of 𝑐, which we can substitute into our equation for the perimeter of the triangle. Let’s clear some space. 𝑎 equals 𝑐 over sin 40. And 𝑏 equals 𝑐 sin 50 over sin 40. Which if we replace in our original equation for the perimeter gives us 𝑐 over sin 40 plus 𝑐 sin 50 over sin 40 plus 𝑐 equals 49. This looks really nasty. But we can factorize by taking out a factor of 𝑐. That gives us 𝑐 multiplied by one over sin 40 plus sin 50 over sin 40 plus one is equal to 49.

If we type the expression inside the brackets into our calculator, we get 3.74747 and so on. We won’t round this number just yet. In fact, we’ll use its exact form in the next stage of our calculation. To solve this equation for 𝑐, we can divide through by this 3.74. And doing so, we get that 𝑐 is equal to 13.075 and so on. Rounding this correct to two decimal places and we get that the length of the shortest side, 𝑐, is 13.08 centimeters.