Question Video: Using the Sine Rule to Calculate an Unknown Length in a Triangle Mathematics

Video Transcript

In triangle 𝐴𝐵𝐶, 𝐴𝐶 is equal to 97 meters, the measure of the angle 𝐵𝐴𝐶 is equal to 101 degrees, and the measure of the angle 𝐴𝐶𝐵 is equal to 53 degrees. Determine the length of 𝐴𝐵 to the nearest meter.

It can be really useful to sketch a diagram in these sorts of scenarios. It’ll allow you to identify the type of question it is and what you will need to use to be able to solve it. Now whilst your diagram does not need to be to scale, it should be roughly in proportion to prevent any mistakes in your calculations.

Here, we have a non-right-angled triangle with two angles and one side given. Notice we don’t have any matching angle and side pairs which is necessary for us to be able to use the law of sines. We do know, however, that we don’t need to use the law of cosines as we simply don’t have enough sides labelled.

We’ll need to use the fact that angles in a triangle add to 180 degrees to work out the measure of the angle 𝐴𝐵𝐶. We can subtract the sum of the given angles from 180 degrees to get the measure of the angle 𝐴𝐵𝐶 to be 26 degrees. Now that we have the measure of the angle at 𝐵, we can use the law of sines. Remember we don’t actually need to use all three parts of this equation.

Labelling our triangle and remembering that the side opposite an angle is given by its lowercase counterpart, we can see that we only need 𝑏 over sin 𝐵 is equal to 𝑐 over sin 𝐶. Substituting our given values into this formula, we get 97 over sin 26 is equal to 𝑐 over sin 53.

We can solve this equation by multiplying both sides by sin 53 to give us sin 53 multiplied by 97 over sin 26 is equal 𝑐. Popping this into our calculator, we get 𝑐 to be 176.717.

The length of 𝐴𝐵 is 177 meters correct to the nearest meter.