Triangles 𝐴𝐵𝐶 and 𝐶𝐷𝐸 are right-angled triangles and 𝐴𝐶 is equal to 𝐶𝐷. Calculate the length of 𝐶𝐸. Give your answer accurate to one decimal place.
Whenever we see right-angled triangles, we should instantly be thinking that we might need to use Pythagoras’s theorem or trigonometry. In fact, in this question, we’re going to use both.
We want to find the length of 𝐶𝐸; that’s the pink line in this triangle. We know that it is a right-angled triangle and we know the measure of one of its other angles. In order to work out the length of 𝐶𝐸 there, we would need another length in this triangle. We are told though that 𝐴𝐶 is equal to 𝐶𝐷. So if we can work out the length of 𝐴𝐶 in triangle 𝐴𝐵𝐶, we’ll have everything we need in triangle 𝐶𝐷𝐸 to work out the length of 𝐶𝐸.
Triangle 𝐴𝐵𝐶 is right-angled and we know the length of two of its sides. We’re going to use Pythagoras’s theorem to find the length of the side 𝐴𝐶. Remember Pythagoras’s theorem says that the square of the hypotenuse is equal to the sum of the squares of the shorter two sides — that sometimes written as 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the hypotenuse, the longest side of our triangle.
We can find the hypotenuse by looking for the side that’s directly opposite the right angle — in this case, that’s the side 𝐴𝐶. Its two shorter sides are four centimetres and 3.5 centimetres in length. We can substitute what we know about this triangle into our formula for Pythagoras’s theorem. That gives us four squared plus 3.5 squared is equal to 𝑐 squared.
We can evaluate these square numbers. Four squared is 16 and 3.5 squared is 12.25. 16 plus 12.25 is 28.25. So 𝑐 squared is equal to 28.25. We can solve this equation by finding the square root. And that gives us 𝑐 is equal to 5.315. These measurements are in centimetres. I’ve added 5.315 to our diagram though I have left it in unrounded form. Where possible, we should try to avoid rounding our numbers until the very end.
Since 𝐴𝐶 is equal to 𝐶𝐷, we can add the measurement for 𝐶𝐷 as 5.315 centimetres. 𝐶𝐷𝐸 is a right-angled triangle, for which we now know the length of one of its sides and the measure of one of its angles. We can use trigonometry. So we’re going to label the triangle.
The hypotenuse is the longest side of the triangle. Once again, it’s the side opposite the right angle. The side opposite the given angle is the opposite. And the other side is the adjacent; that’s the one next to the given angle. We know the length of the opposite and we’re trying to find the length of the adjacent. We can use the mnemonic SOHCAHTOA.
We’re looking to find the adjacent side in the triangle and we know the length of the opposite side. This means we need to use the tan ratio. tan of 𝜃 is equal to opposite over adjacent. 𝜃 is the angle; in this case, that’s 41 degrees. The length of the opposite side is 5.315. Let’s call 𝐶𝐸 𝑥 for now.
We need to solve this equation for 𝑥. The first thing we’re going to do is multiply both sides of the equation by 𝑥 and that gives us 𝑥 multiplied by tan of 41 is equal to 5.315. To solve, we need to divide both sides by tan of 41. And that gives us that 𝑥 is equal to 5.315 all divided by tan of 41. That’s 6.114. Correct to one decimal place, the length of 𝐶𝐸 is 6.1 centimetres.
Now, let’s look at an alternative way to find the length of 𝐶𝐸. Once we’ve identified that we need to use tan, we can use a triangle. We’re trying to calculate the length of the adjacent side. So we cover this up and it tells us what to do with the opposite and the tan.
This line in the triangle tells us to divide. So 𝑥 is going to be equal to 5.315 divided by tan of 41. That’s 6.114. Correct to one decimal place once again, we calculated the length of 𝐶𝐸 to be 6.1 centimetres.