In the given figures 𝐴𝐵 equals six, 𝐴𝐷 equals three, and the measure of angle 𝐴𝐶𝐵 is 33 degrees. Calculate the length of 𝐵𝐶. Give your answer to two decimal places.
First, let’s transfer the written information in the question onto the diagram. Now let’s have a closer look at this diagram. It consists of two right-angled triangles which share a common side, 𝐵𝐷. In the triangle on the left, we’ve have been given the lengths of two sides. In the triangle on the right, we’ve been given the measure of one angle.
The side we have been asked to calculate, 𝐵𝐶, is in the second triangle in which we currently only have one piece of information. In order to find this side, we’re going to need at least two pieces of information in the triangle to which it belongs. So we’re going to need to use the fact that the triangles share the side 𝐵𝐷.
Let’s begin in triangle 𝐴𝐵𝐷. As we’ve already said, this is a right-angled triangle. And we know the lengths of two of its sides. Therefore, we can apply the Pythagorean theorem in order to calculate the length of the third side. The Pythagorean theorem tells us that in this triangle, 𝐵𝐷 squared plus three squared is equal to six squared. And this is an equation that we can solve in order to find the length of 𝐵𝐷.
Evaluating three squared and six squared tells us that 𝐵𝐷 squared plus nine is equal to 36. Subtracting nine from both sides of this equation gives 𝐵𝐷 squared is equal to 27. Finally for this stage, square rooting both sides of the equation tells us that 𝐵𝐷 is equal to the square root of 27.
Now I could evaluate this as a decimal but it would be a value that needed rounding and therefore an inexact value. So I’m going to keep it as a surd for now. So now we know the length of 𝐵𝐷. Let’s turn our attention to the second triangle, the triangle on the right. Apart from the right angle, we now have one known angle and one known side. And we’re looking to calculate the length of a second side in this triangle.
We can use trigonometry in order to do this. The first step is to label the three sides of the triangle in relation to the angle of 33 degrees. So 𝐵𝐶 is the hypotenuse, 𝐶𝐷 is the adjacent, and 𝐵𝐷 is the opposite. Remember it’s 𝐵𝐶 that we’re looking to calculate. We know the length of the opposite side, 𝐵𝐷. And we’re looking to calculate the hypotenuse.
So if we recall the acronym SOHCAHTOA, this tells us that we need to use the sine ratio. The sine ratio is defined as sine of an angle 𝜃 is equal to the opposite divided by the hypotenuse. Now let’s substitute the values for this triangle. We have that sine of 33 degrees is equal to root 27 over 𝐵𝐶.
This is why I kept that value as a square root rather than evaluating as a decimal, because I knew I was going to use it in the next stage of the calculation. Now we want to solve this equation in order to find the value of 𝐵𝐶. So I’m going to multiply by 𝐵𝐶, first of all, as it’s currently in the denominator of a fraction.
This gives me 𝐵𝐶 multiplied by sin 33 degrees is equal to root 27. Next, I need to divide both sides of the equation by sine of 33 degrees, which is just a number. So we have that 𝐵𝐶 is equal to root 27 over sin 33 degrees.
Now I can evaluate this using my calculator. And this is actually the first stage in this question that I’ve needed to use a calculator. This gives a value of 9.54054 and the decimal then continues. The question has asked us to give our answer to two decimal places. So we need to round this value.
So we have that the length of 𝐵𝐶 to two decimal places is 9.54. And there are no units with this as we weren’t given any units in the original question.