Video: Finding the Lengths of Proportional Line Segments between Parallel Lines

Given that ๐ด๐ถ = 7.5 cm, ๐ต๐ท = 14 cm, ๐น๐‘Œ = 25.2 cm, and ๐น๐พ = 42 cm, determine the lengths of lines ๐ถ๐‘‹ and ๐ท๐น.

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Video Transcript

Given that ๐ด๐ถ equals 7.5 centimetres, ๐ต๐ท equals 14 centimetres, ๐น๐‘Œ equals 25.2 centimetres, and ๐น๐พ equals 42 centimetres, determine the lengths of ๐ถ๐‘‹ and ๐ท๐น.

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 weโ€™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 weโ€™re using is one of our angle properties, so an angle property of parallel lines, to help us work out that theyโ€™re the same. And that is that we have corresponding angles. Sometimes thereโ€™s F-angles because itโ€™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 weโ€™re gonna do is weโ€™re gonna use this property to help us figure out the lengths that weโ€™re looking for. Well, first of all, weโ€™re gonna use three lengths that weโ€™ve been given. So weโ€™ve got ๐ต๐ท is equal to 14 centimetres. ๐ด๐ถ equals 7.5 centimetres. Then weโ€™ve got ๐น๐‘Œ equals 25.2 centimetres. So therefore, what we can do is we can use these distances to help us calculate the distance between ๐ธ and ๐‘‹. Because as weโ€™ve already said theyโ€™re gonna have equal ratios of the lengths of corresponding sides because theyโ€™re similar shapes, so the shapes ๐ต๐ท๐ด๐ถ and ๐น๐‘Œ๐ธ๐‘‹.

So we can say that ๐น๐‘Œ over ๐ต๐ท, because theyโ€™re corresponding sides, are gonna be equal to ๐ธ๐‘‹ over ๐ด๐ถ. So therefore, 25.2 over 14 is gonna be equal to ๐ธ๐‘‹ over 7.5. So what weโ€™re gonna get is 1.8 is equal to ๐ธ๐‘‹ over 7.5. Itโ€™s worth pointing out that what weโ€™ve actually done here is found the scale factor of enlargement. And thatโ€™s because the scale factor is equal to the new length over the original length. So we had ๐น๐‘Œ over ๐ต๐ท.

Okay, great. So now we know that 1.8 is equal to ๐ธ๐‘‹ over 7.5. All we need to do is multiply by both sides by 7.5 to give us what ๐ธ๐‘‹ is going to be. And this gives the length of 13.5 centimetres. So great, we know that ๐ธ๐‘‹ 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 ๐ธ๐‘. And thatโ€™s because we know the distance ๐น๐พ is 42.

So then again, what we can say this time is that ๐น๐พ over ๐ต๐ท โ€” and thatโ€™s because theyโ€™re corresponding sides โ€” is equal to ๐ธ๐‘ over ๐ด๐ถ. So what weโ€™re gonna do when we work out ๐น๐พ over ๐ต๐ท is in fact gonna work out our scale factor once again. So weโ€™re gonna have 42 over 14 is equal to ๐ธ๐‘ over 7.5. So thatโ€™s gonna give us three is equal to ๐ธ๐‘ over 7.5.

So then we multiply both sides by 7.5. So when we do this, weโ€™re gonna get 22.5 centimetres equal to ๐ธ๐‘. So again, Iโ€™ve marked this on our diagram. So great, we found ๐ธ๐‘‹ and ๐ธ๐‘. But why is this useful?

Well, this is useful because what we can do now is work out segment ๐ถ๐ธ. And thatโ€™s because ๐ถ๐ธ is gonna be equal to 45 minus ๐ธ๐‘. So ๐ถ๐ธ is gonna be equal to 45 minus 22.5. Well, ๐ถ๐ธ is also gonna be equal to 22.5 centimetres. So thatโ€™s great. So what we can do now is get on and work out the length of the line segment ๐ถ๐‘‹ and ๐ท๐น and solve the problem.

Well, first of all, the line segment ๐ถ๐‘‹ is gonna be equal to ๐ถ๐ธ plus ๐ธ๐‘‹. So this is gonna be equal to 22.5 plus 13.5. So therefore, weโ€™ve solved the first part of the problem. We know that the line segment ๐ถ๐‘‹ is gonna be equal to 36 centimetres.

Okay, great. Now we can move on to ๐ท๐น. Well, what we can do now is use the same method that we used before. We can say that ๐ถ๐ธ over ๐ด๐ถ is equal to ๐ท๐น over ๐ต๐ท cause these are corresponding sides. So therefore, we can say that 22.5 over 7.5 is equal to ๐ท๐น over 14. So therefore, we can say three is equal to ๐ท๐น over 14. And then if we multiply both sides by three, weโ€™re gonna get 42 is equal to ๐ท๐น. So therefore, we can say that the line segment ๐ท๐น is equal to 42 centimetres. So weโ€™ve solved the problem.

Itโ€™s worth noting that we also shouldโ€™ve known that this segment was going to be 42 centimetres because we knew that ๐ธ๐‘ was equal to 22.5 and ๐ถ๐ธ were equal to 22.5. So therefore, these are the same. So their corresponding other side should be the same. So therefore, ๐ท๐น was gonna be equal to ๐น๐พ, which was equal to 42 centimetres.

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