# Question Video: Using Properties of Parallel Lines to Solve a Problem Mathematics • 11th Grade

If π΅π = 22 cm, π΄π = 30 cm, and (π΄π + π΄π)/(π΄π΅ + π΄πΆ) = 10/21, find the length of the line segment πΆπ.

03:49

### Video Transcript

If π΅π equals 22 centimeters, π΄π equals 30 centimeters, and π΄π plus π΄π over π΄π΅ plus π΄πΆ equals 10 over 21, find the length of the line segment πΆπ.

So weβre told that π΄π plus π΄π over π΄π΅ plus π΄πΆ is equal to 10 over 21. But how is this going to be relevant? Well, before weβre gonna have a look at the relevance of this, letβs have a look at some of the properties weβve got with our shape. First of all here, weβve got two parallel lines. So what we can see is that the parallel lines actually break our triangle into two. So youβve got two triangles, triangle π΄ππ and triangle π΄π΅πΆ.

And because weβre looking at these parallel lines, it means that weβve got this line that transverses them, which is π΄πΆ. And we can see that we have corresponding angles, and we had shown it here in orange. And this means that also if we take a look at the other side, we also have another pair of corresponding angles, here, an angle at π and an angle at π΅. And we can see once again these are going to be the same. And we can also see that our two triangles have a shared angle at π΄, so this is going to be the same in both triangle as well.

So therefore, we can say that triangle π΄π΅πΆ is similar to triangle π΄ππ. And similar means that they are in the same proportion, but not necessarily the same size. So, in fact, one is a dilation or an enlargement of the other. And the reason we know theyβre similar is because we have used the angle-angle or angle-angle-angle proof. And we can do this because we know that angle π΄ is equal to angle π΄, angle π is equal to angle πΆ, and angle π is equal to angle π΅.

Okay, great. But what does this mean? How does it help us? Well, it helps us because as we said before, in similar triangles, corresponding sides are always proportional. So therefore, we can see that π΄π and π΄π΅ are corresponding and π΄π and π΄πΆ are corresponding. And we can see that π΄π plus π΄π over π΄π΅ plus π΄πΆ is equal to 10 over 21. Then, therefore, π΄π over π΄πΆ must also be equal to 10 over 21 because as we said corresponding sides are always proportional. And weβve got a proportion shown as 10 over 21.

So then, if we multiply both sides of our equation here by π΄πΆ and 21, weβre gonna get 21π΄π equals 10π΄πΆ. So therefore, π΄πΆ is gonna be equal to 21 over 10 π΄π.

Okay, great. But how is this useful? Well, itβs useful because we know what π΄π is, because π΄π is equal to 30 centimeters. So therefore, we can say that π΄πΆ is equal to 21 over 10 multiplied by 30. Well then, what we can do, rather than having to multiply 21 over 10 by 30, we can divide both the numerator and denominator by 10. So what weβre gonna have is 21 multiplied by three over one. So therefore, we can say that π΄πΆ is equal to 63 centimeters.

Okay, great. So how is this gonna help us now? Well, what weβre looking for is to find the line segment πΆπ. So weβve now got π΄πΆ. Weβve also got π΄π. So we can use them both together to find πΆπ. Well, line segment πΆπ is gonna be equal to π΄πΆ minus π΄π, which is gonna be equal to 63 minus 30, which will give us a final answer of 33.

So therefore, we can say that the line segment πΆπ will be 33 centimeters long.

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