Question Video: Finding the Unknown Lengths in a Triangle given the Other Sides’ Lengths Using the Relations of Parallel Lines Mathematics

Video Transcript

Given that 𝐴𝐵𝐶𝐷 is a parallelogram, find the length of line segment 𝑌𝑍.

In this question, we need to find this line segment 𝑌𝑍. You may think that this is going to be quite difficult. After all, we’re only given the length of one side in this parallelogram, 𝐴𝐵. We do, however, have two sets of congruent length. And we know that 𝐴𝐵𝐶𝐷 is a parallelogram. So let’s see if we can work out any proportionalities or any lengths. We may even be able to apply the side splitter theorem here. This theorem tells us that if a line is parallel to one side of a triangle and intersects the other two sides, then the line divides the two sides proportionally.

However, in the case of this diagram, if we look at the triangle 𝐷𝑋𝐶, we aren’t given the information that 𝑌𝑍 is parallel to 𝐷𝐶. But we can remember that the converse of the side splitter theorem also holds. That is, if there is a line which splits two sides of a triangle proportionally, then that line is parallel to the remaining side. So let’s look at what’s happening in this triangle 𝐷𝑋𝐶. From the markings on the diagram, we can see that 𝐷𝑌 is exactly equal to 𝑌𝑋. And so we could say that 𝐷𝑌 is half of the whole length of 𝐷𝑋. In the same way, 𝐶𝑍 is equal to 𝑍𝑋. So 𝐶𝑍 is half of the whole length of 𝐶𝑋. So the length of 𝑍𝑌 has split both of the other two sides into two equal pieces.

As an aside, however, the side splitter theorem doesn’t necessarily divide the two other sides into two equal lengths. For example, in this triangle below, there is a line parallel to one of the sides of the triangle. The two other sides are split proportionally, both in the ratio one to three. But now let’s return to this parallelogram. Because we have the two sides 𝑋𝐷 and 𝑋𝐶 in the larger triangle divided into equal proportions, then we have that the line segment 𝐷𝐶 is parallel to the line segment 𝑌𝑍. We can also note that triangle 𝑋𝑌𝑍 is similar to triangle 𝑋𝐷𝐶.

Since 𝐴𝐵𝐶𝐷 is a parallelogram, we know that there are two opposite sides of equal lengths. So the length of line segment 𝐷𝐶 is also 134.9 centimeters. Now, let’s see if we can work out any unknown lengths and let’s denote this length of 𝑋𝑌 as 𝑋 centimeters. Therefore, since 𝐷𝑌 is also 𝑋 centimeters, then the whole length of the line segment 𝐷𝑋 would be two 𝑋 centimeters. We can then use the fact that these triangles are similar to write a proportionality statement. And let’s include one of the sides of which we know its actual length. And so 𝑋𝑌 over 𝑋𝐷 must be equal to the proportionality 𝑌𝑍 over 𝐷𝐶.

We defined 𝑋𝑌 as 𝑋 and 𝑋𝐷 as two 𝑋. And then on the right-hand side, we don’t yet know 𝑌𝑍, but we do know that 𝐷𝐶 is 134.9 centimeters. We have a few unknowns here. But don’t forget that on the left-hand side, if we have effectively one 𝑋 over two 𝑋, then that will simplify to one-half. The right-hand side will stay the same. But now we can solve for 𝑌𝑍. This means that two times 𝑌𝑍 is equal to 134.9. And then we divide both sides by two. We can therefore give the answer that the length of the line segment 𝑌𝑍 is 67.45 centimeters.