Question Video: The Sine Rule Mathematics

Video Transcript

𝐴𝐵𝐶 is a triangle, where two sin 𝐴 equals three sin 𝐵 equals four sin 𝐶 and the perimeter is 169 centimetres. Find the values of 𝑎 and 𝑐, giving the answer to the nearest centimetre.

We know the relationships between the sin of 𝐴, sin of 𝐵, and sin of 𝐶. Notice that this equation looks a little bit like the law of sines. If we compare this to the formula for the law of sines, we can see that the coefficient for sin 𝐴, sin 𝐵, and sin 𝐶 in the general form are all fractions.

In order to make our equation look like this, we’ll need to divide through by some common multiple of two, three, and four. The lowest common multiple of these three numbers is 12. So let’s begin by dividing through our entire equation by 12. And simplified, we get two sin 𝐴 over 12 equals three sin 𝐵 over 12, which equals four sin 𝐶 over 12.

Let’s simplify each of these fractions. We can divide through this first fraction by a common factor of two. That gives us sin 𝐴 over six. We can divide through the second fraction by a common factor of three. That gives us sin 𝐵 over four. And finally, dividing through by a common factor of four for the third fraction gives us sin 𝐶 over three.

sin 𝐴 over six is equal to sin 𝐵 over four, which is equal to sin 𝐶 over three. The denominators of these fractions correspond to the length of 𝑎, 𝑏, and 𝑐 in our formula for the law of sines. This means then that the ratio of the length of the sides of the triangle 𝑎 to 𝑏 to 𝑐 can be written as six to four to three.

Now, we know that the perimeter of the shape is 169 centimetres, where the perimeter is the sum total of the length of the triangles. To calculate the individual length of the triangle then, we’ll need to share 169 centimetres in the ratio of six to four to three. Six plus four plus three is 13. There are a total of 13 parts.

𝑎 is worth six of these. 𝑎 must be six thirteenths of the perimeter. That’s six thirteenths of 169. Six thirteenths of 169 is 78. So 𝑎 must be 78 centimetres. 𝑐 is worth three of those 13 parts. It’s three thirteenths of the perimeter. Three thirteenths of 169 is 39. So 𝑐 must be 39 centimetres.

We can check our answer by calculating the length of 𝑏, which is four thirteenths of the perimeter. Four thirteenths of 169 is 52. So 𝑏 is 52 centimetres. If we add these three lengths, we get 169, which is the perimeter. This gives us a strong indication that the calculations we’ve performed are correct.

𝑎 is 78 centimetres and 𝑐 is 39 centimetres.