Given that 𝐴𝐶 equals 7.5 centimetres, 𝐵𝐷 equals 14 centimetres, 𝐹𝑌 equals 25.2 centimetres, and 𝐹𝐾 equals 42 centimetres, determine the lengths of 𝐶𝑋 and 𝐷𝐹.
Well, seeing that we have a set of parallel lines that make up our shape, because what we have is a series of trapezia. Then we can say that the angles that we’ve marked are gonna be equal. So the pink angles are gonna be equal. The orange angles are gonna be equal. The blue angles are gonna be equal. And so are the green angles.
And what we’re using is one of our angle properties, so an angle property of parallel lines, to help us work out that they’re the same. And that is that we have corresponding angles. Sometimes there’s F-angles because it’s like they make an F when you look at them together in pairs. Okay, great. But what does this tell us? How does this help?
Well, what this tells us is that our trapezia within the larger trapezia are all similar trapezia or similar shapes. And why is this useful? When we have mathematically similar shapes, what they are in fact is enlargements of each other. So therefore, we can say that two similar shapes have equal ratios of the lengths of their corresponding sides.
What we’re gonna do is we’re gonna use this property to help us figure out the lengths that we’re looking for. Well, first of all, we’re gonna use three lengths that we’ve been given. So we’ve got 𝐵𝐷 is equal to 14 centimetres. 𝐴𝐶 equals 7.5 centimetres. Then we’ve got 𝐹𝑌 equals 25.2 centimetres. So therefore, what we can do is we can use these distances to help us calculate the distance between 𝐸 and 𝑋. Because as we’ve already said they’re gonna have equal ratios of the lengths of corresponding sides because they’re similar shapes, so the shapes 𝐵𝐷𝐴𝐶 and 𝐹𝑌𝐸𝑋.
So we can say that 𝐹𝑌 over 𝐵𝐷, because they’re corresponding sides, are gonna be equal to 𝐸𝑋 over 𝐴𝐶. So therefore, 25.2 over 14 is gonna be equal to 𝐸𝑋 over 7.5. So what we’re gonna get is 1.8 is equal to 𝐸𝑋 over 7.5. It’s worth pointing out that what we’ve actually done here is found the scale factor of enlargement. And that’s because the scale factor is equal to the new length over the original length. So we had 𝐹𝑌 over 𝐵𝐷.
Okay, great. So now we know that 1.8 is equal to 𝐸𝑋 over 7.5. All we need to do is multiply by both sides by 7.5 to give us what 𝐸𝑋 is going to be. And this gives the length of 13.5 centimetres. So great, we know that 𝐸𝑋 is equal to 13.5 centimetres. So now what we can do is we can use a very similar method to help us work out the distance 𝐸𝑍. And that’s because we know the distance 𝐹𝐾 is 42.
So then again, what we can say this time is that 𝐹𝐾 over 𝐵𝐷 — and that’s because they’re corresponding sides — is equal to 𝐸𝑍 over 𝐴𝐶. So what we’re gonna do when we work out 𝐹𝐾 over 𝐵𝐷 is in fact gonna work out our scale factor once again. So we’re gonna have 42 over 14 is equal to 𝐸𝑍 over 7.5. So that’s gonna give us three is equal to 𝐸𝑍 over 7.5.
So then we multiply both sides by 7.5. So when we do this, we’re gonna get 22.5 centimetres equal to 𝐸𝑍. So again, I’ve marked this on our diagram. So great, we found 𝐸𝑋 and 𝐸𝑍. But why is this useful?
Well, this is useful because what we can do now is work out segment 𝐶𝐸. And that’s because 𝐶𝐸 is gonna be equal to 45 minus 𝐸𝑍. So 𝐶𝐸 is gonna be equal to 45 minus 22.5. Well, 𝐶𝐸 is also gonna be equal to 22.5 centimetres. So that’s great. So what we can do now is get on and work out the length of the line segment 𝐶𝑋 and 𝐷𝐹 and solve the problem.
Well, first of all, the line segment 𝐶𝑋 is gonna be equal to 𝐶𝐸 plus 𝐸𝑋. So this is gonna be equal to 22.5 plus 13.5. So therefore, we’ve solved the first part of the problem. We know that the line segment 𝐶𝑋 is gonna be equal to 36 centimetres.
Okay, great. Now we can move on to 𝐷𝐹. Well, what we can do now is use the same method that we used before. We can say that 𝐶𝐸 over 𝐴𝐶 is equal to 𝐷𝐹 over 𝐵𝐷 cause these are corresponding sides. So therefore, we can say that 22.5 over 7.5 is equal to 𝐷𝐹 over 14. So therefore, we can say three is equal to 𝐷𝐹 over 14. And then if we multiply both sides by three, we’re gonna get 42 is equal to 𝐷𝐹. So therefore, we can say that the line segment 𝐷𝐹 is equal to 42 centimetres. So we’ve solved the problem.
It’s worth noting that we also should’ve known that this segment was going to be 42 centimetres because we knew that 𝐸𝑍 was equal to 22.5 and 𝐶𝐸 were equal to 22.5. So therefore, these are the same. So their corresponding other side should be the same. So therefore, 𝐷𝐹 was gonna be equal to 𝐹𝐾, which was equal to 42 centimetres.