Given that 𝐷𝐸 is 74 meters, 𝐸𝐵 is 32 meters, and 𝐸𝐴 is 48 meters, find the length of 𝐶𝐴.
First, let’s add the information we’ve been given in the question onto the diagram. So we have two right-angled triangles in which we’ve been given various lengths. And we’re asked to find the length of 𝐶𝐴 which is, in fact, the length of the two hypotenuses of the triangle joined together, 𝐶𝐸 and 𝐸𝐴.
So in order to answer the question, we’re going to need to work out the length of 𝐶𝐸 first of all. So let’s think about how we’re going to do this. We’ve been given two triangles, both right-angled triangles. And as I said, we’ve been given lengths in each of them.
This suggests that we might want to approach this problem using similar triangles. We can’t just assume that the two triangles are similar. We need to show it first. In order for two triangles to be similar, they need to have all three angles the same. Both triangles have a right angle. So first of all we can say that angle 𝐶𝐷𝐸 is congruent to angle 𝐴𝐵𝐸.
Let’s look at the pair of angles marked in green. This pair of angles are vertically opposite one another, as they’re formed by a pair of intersecting straight lines. Therefore, these two angles are equal. So we have that angle 𝐷𝐸𝐶 is congruent to angle 𝐵𝐸𝐴. Now it follows automatically that the third angle in these two triangles is also equal because the angle sum in a triangle is always 180 degrees.
And if the other two angles in each triangle are equal, then the third must also be the same. For this reason, it’s actually sufficient to show that two of the angles in a pair of triangles are the same in order to conclude that the two triangles are similar. So we can conclude that triangle 𝐶𝐷𝐸 is similar to triangle 𝐴𝐵𝐸.
Now how does this help us with answering the question? Well, remember that if two triangles are similar, then they have proportional side lengths. This means that if I divide a side in the larger triangle by a corresponding side in the smaller triangle, then I always get the same ratio. Let’s substitute in the values for the sides we know.
We have 74 divided by 32 for 𝐷𝐸 and 𝐵𝐸, the horizontal side of the two triangles. And then we have 𝐶𝐸 divided by 48 for the two hypotenuses. This gives an equation that we can solve in order to find 𝐶𝐸. We need to multiply both sides of the equation by 48. So 𝐶𝐸 is 48 multiplied by 74 over 32, which is 111.
Now we haven’t finished the question yet because, remember, we weren’t asked just to calculate 𝐶𝐸, we were asked to calculate 𝐶𝐴, which is this full length here, the sum of 𝐶𝐸 and 𝐸𝐴. But we know both of these lengths. So we just need to add them together.
We have that 𝐶𝐴 is equal to 111 plus 48, which gives us our answer of 159 meters. Remember, we used similar triangles in this question. But we couldn’t just assume the two triangles were similar. We had to prove it first by giving the reasoning for why their angles were equal.