𝐴𝐵𝐶 is a right-angled triangle at 𝐵. The point 𝐷 lies on the vector 𝐵𝐶, where 𝐶𝐷 is equal to 17 centimetres, the measure of the angle 𝐴𝐷𝐶 is 46 degrees, and the measure of the angle 𝐶𝐴𝐷 is 24 degrees. Find the length of 𝐴𝐵, giving your answer to the nearest centimetre.
The clue here is that the point 𝐷 lies on the vector 𝐵𝐶. There’s actually no way the point can lie between 𝐵 and 𝐶 and still be 46 degrees. Let’s sketch this out and see what it looks like. Notice that we know two of the angles in the triangle 𝐴𝐶𝐷 and the length of one of its sides. This means we can use the law of sines to calculate the length of the side 𝐴𝐶: that’s the side shared by both of the triangles.
We know that we need to use the law of sines over the law of cosines since that law requires at least two known sides. Now, we can use either of these forms for the sine rule. Since we’re trying to find a side though, it’s sensible to use the first form in order to minimize the amount of rearranging we need to do.
The second form for the law of sines is better when we’re trying to calculate the measure of one of the angles. Let’s begin by labelling the sides of this triangle. The side 𝑎 sits directly opposite the angle 𝐴, the side 𝑐 sits directly opposite angle 𝐶, and the side 𝑑 sits directly opposite the angle 𝐷.
The rule will change slightly to 𝑎 over sin 𝐴 equals 𝑐 over sin 𝐶 equals 𝑑 over sin 𝐷 to account for the names of the angles in our triangle. Since we’re using the side 𝑎 and trying to find the side 𝑑, we’ll use these two parts of the equation.
Substituting the relevant values into our equation gives us 17 over sin of 24 equals 𝑑 over sin of 46. We’ll then solve this equation by multiplying both sides by sin of 46. And that gives us that 𝑑 is equal to 17 over sin of 24 multiplied by sin of 46. Popping this into our calculator, we get a value of 30.065 and so on. We won’t round this answer just yet in order to prevent any mistakes from rounding too early.
We’ve calculated the length of the side 𝐴𝐶. That’s the length of the hypotenuse of this right-angled triangle. We can use right angle trigonometry to find the length 𝐴𝐵, which I’ve labelled as 𝑥. Before we do though, we can calculate the measure of the acute angle at 𝐶. Remember this angle is equal to the sum of the angles at 𝐴 and 𝐷. The exterior angle in a triangle will always be equal to the sum of the two interior opposite angles. That’s 70 degrees.
Now, let’s label the triangle 𝐴𝐵𝐶 using the conventions for right angle trigonometry. The length 𝐴𝐶 is the hypotenuse of the triangle. And we’re trying to find the length 𝐴𝐵, which is the opposite side. It’s the side opposite the angle we just calculated. In this case then, we can use the sine ratio to help us find the length that we’ve labelled 𝑥.
Substituting what we know from our triangle into this formula gives us sin of 70 equals 𝑥 over 30.065. Then, we can solve this equation by multiplying both sides by our nonrounded number 30.065. That’s 𝑥 is equal to sin of 70 multiplied by 30.06 and so on. And when we type it into our calculator, we get a value of 28.252.
Correct to the nearest centimetre, the length of 𝐴𝐵 is 28 centimetres.