If 𝐵𝑋 equals 22 centimeters, 𝐴𝑌 equals 30 centimeters, and 𝐴𝑋 plus 𝐴𝑌 over 𝐴𝐵 plus 𝐴𝐶 equals 10 over 21, find the length of the line segment 𝐶𝑌.
So we’re told that 𝐴𝑋 plus 𝐴𝑌 over 𝐴𝐵 plus 𝐴𝐶 is equal to 10 over 21. But how is this going to be relevant? Well, before we’re gonna have a look at the relevance of this, let’s have a look at some of the properties we’ve got with our shape. First of all here, we’ve got two parallel lines. So what we can see is that the parallel lines actually break our triangle into two. So you’ve got two triangles, triangle 𝐴𝑋𝑌 and triangle 𝐴𝐵𝐶.
And because we’re looking at these parallel lines, it means that we’ve got this line that transverses them, which is 𝐴𝐶. 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 𝑋 and an angle at 𝐵. 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 𝐴, so this is going to be the same in both triangle as well.
So therefore, we can say that triangle 𝐴𝐵𝐶 is similar to triangle 𝐴𝑋𝑌. 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 they’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 𝐴 is equal to angle 𝐴, angle 𝑌 is equal to angle 𝐶, and angle 𝑋 is equal to angle 𝐵.
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 𝐴𝑋 and 𝐴𝐵 are corresponding and 𝐴𝑌 and 𝐴𝐶 are corresponding. And we can see that 𝐴𝑋 plus 𝐴𝑌 over 𝐴𝐵 plus 𝐴𝐶 is equal to 10 over 21. Then, therefore, 𝐴𝑌 over 𝐴𝐶 must also be equal to 10 over 21 because as we said corresponding sides are always proportional. And we’ve got a proportion shown as 10 over 21.
So then, if we multiply both sides of our equation here by 𝐴𝐶 and 21, we’re gonna get 21𝐴𝑌 equals 10𝐴𝐶. So therefore, 𝐴𝐶 is gonna be equal to 21 over 10 𝐴𝑌.
Okay, great. But how is this useful? Well, it’s useful because we know what 𝐴𝑌 is, because 𝐴𝑌 is equal to 30 centimeters. So therefore, we can say that 𝐴𝐶 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 we’re gonna have is 21 multiplied by three over one. So therefore, we can say that 𝐴𝐶 is equal to 63 centimeters.
Okay, great. So how is this gonna help us now? Well, what we’re looking for is to find the line segment 𝐶𝑌. So we’ve now got 𝐴𝐶. We’ve also got 𝐴𝑌. So we can use them both together to find 𝐶𝑌. Well, line segment 𝐶𝑌 is gonna be equal to 𝐴𝐶 minus 𝐴𝑌, 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 𝐶𝑌 will be 33 centimeters long.