### Video Transcript

Given that ๐ด๐ถ equals 7.5 centimetres, ๐ต๐ท equals 14 centimetres, ๐น๐ equals 25.2 centimetres, and ๐น๐พ equals 42 centimetres, determine the lengths of ๐ถ๐ and ๐ท๐น.

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 weโve 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 weโre using is one of our angle properties, so an angle property of parallel lines, to help us work out that theyโre the same. And that is that we have corresponding angles. Sometimes thereโs F-angles because itโs 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 weโre gonna do is weโre gonna use this property to help us figure out the lengths that weโre looking for. Well, first of all, weโre gonna use three lengths that weโve been given. So weโve got ๐ต๐ท is equal to 14 centimetres. ๐ด๐ถ equals 7.5 centimetres. Then weโve got ๐น๐ equals 25.2 centimetres. So therefore, what we can do is we can use these distances to help us calculate the distance between ๐ธ and ๐. Because as weโve already said theyโre gonna have equal ratios of the lengths of corresponding sides because theyโre similar shapes, so the shapes ๐ต๐ท๐ด๐ถ and ๐น๐๐ธ๐.

So we can say that ๐น๐ over ๐ต๐ท, because theyโre corresponding sides, are gonna be equal to ๐ธ๐ over ๐ด๐ถ. So therefore, 25.2 over 14 is gonna be equal to ๐ธ๐ over 7.5. So what weโre gonna get is 1.8 is equal to ๐ธ๐ over 7.5. Itโs worth pointing out that what weโve actually done here is found the scale factor of enlargement. And thatโs because the scale factor is equal to the new length over the original length. So we had ๐น๐ over ๐ต๐ท.

Okay, great. So now we know that 1.8 is equal to ๐ธ๐ over 7.5. All we need to do is multiply by both sides by 7.5 to give us what ๐ธ๐ is going to be. And this gives the length of 13.5 centimetres. So great, we know that ๐ธ๐ 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 ๐ธ๐. And thatโs because we know the distance ๐น๐พ is 42.

So then again, what we can say this time is that ๐น๐พ over ๐ต๐ท โ and thatโs because theyโre corresponding sides โ is equal to ๐ธ๐ over ๐ด๐ถ. So what weโre gonna do when we work out ๐น๐พ over ๐ต๐ท is in fact gonna work out our scale factor once again. So weโre gonna have 42 over 14 is equal to ๐ธ๐ over 7.5. So thatโs gonna give us three is equal to ๐ธ๐ over 7.5.

So then we multiply both sides by 7.5. So when we do this, weโre gonna get 22.5 centimetres equal to ๐ธ๐. So again, Iโve marked this on our diagram. So great, we found ๐ธ๐ and ๐ธ๐. But why is this useful?

Well, this is useful because what we can do now is work out segment ๐ถ๐ธ. And thatโs because ๐ถ๐ธ is gonna be equal to 45 minus ๐ธ๐. So ๐ถ๐ธ is gonna be equal to 45 minus 22.5. Well, ๐ถ๐ธ is also gonna be equal to 22.5 centimetres. So thatโs great. So what we can do now is get on and work out the length of the line segment ๐ถ๐ and ๐ท๐น and solve the problem.

Well, first of all, the line segment ๐ถ๐ is gonna be equal to ๐ถ๐ธ plus ๐ธ๐. So this is gonna be equal to 22.5 plus 13.5. So therefore, weโve solved the first part of the problem. We know that the line segment ๐ถ๐ is gonna be equal to 36 centimetres.

Okay, great. Now we can move on to ๐ท๐น. Well, what we can do now is use the same method that we used before. We can say that ๐ถ๐ธ over ๐ด๐ถ is equal to ๐ท๐น over ๐ต๐ท cause these are corresponding sides. So therefore, we can say that 22.5 over 7.5 is equal to ๐ท๐น over 14. So therefore, we can say three is equal to ๐ท๐น over 14. And then if we multiply both sides by three, weโre gonna get 42 is equal to ๐ท๐น. So therefore, we can say that the line segment ๐ท๐น is equal to 42 centimetres. So weโve solved the problem.

Itโs worth noting that we also shouldโve known that this segment was going to be 42 centimetres because we knew that ๐ธ๐ was equal to 22.5 and ๐ถ๐ธ were equal to 22.5. So therefore, these are the same. So their corresponding other side should be the same. So therefore, ๐ท๐น was gonna be equal to ๐น๐พ, which was equal to 42 centimetres.