### Video Transcript

Given that π΄π΅πΆπ· is a parallelogram, find the length of line segment ππ.

In this question, we need to find this line segment ππ. You may think that this is going to be quite difficult. After all, weβre only given the length of one side in this parallelogram, π΄π΅. We do, however, have two sets of congruent length. And we know that π΄π΅πΆπ· is a parallelogram. So letβs 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 π·ππΆ, we arenβt given the information that ππ is parallel to π·πΆ. 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 letβs look at whatβs happening in this triangle π·ππΆ. From the markings on the diagram, we can see that π·π is exactly equal to ππ. And so we could say that π·π is half of the whole length of π·π. In the same way, πΆπ is equal to ππ. So πΆπ is half of the whole length of πΆπ. So the length of ππ has split both of the other two sides into two equal pieces.

As an aside, however, the side splitter theorem doesnβt 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 letβs return to this parallelogram. Because we have the two sides ππ· and ππΆ in the larger triangle divided into equal proportions, then we have that the line segment π·πΆ is parallel to the line segment ππ. We can also note that triangle πππ is similar to triangle ππ·πΆ.

Since π΄π΅πΆπ· is a parallelogram, we know that there are two opposite sides of equal lengths. So the length of line segment π·πΆ is also 134.9 centimeters. Now, letβs see if we can work out any unknown lengths and letβs denote this length of ππ as π centimeters. Therefore, since π·π is also π centimeters, then the whole length of the line segment π·π would be two π centimeters. We can then use the fact that these triangles are similar to write a proportionality statement. And letβs include one of the sides of which we know its actual length. And so ππ over ππ· must be equal to the proportionality ππ over π·πΆ.

We defined ππ as π and ππ· as two π. And then on the right-hand side, we donβt yet know ππ, but we do know that π·πΆ is 134.9 centimeters. We have a few unknowns here. But donβt forget that on the left-hand side, if we have effectively one π over two π, then that will simplify to one-half. The right-hand side will stay the same. But now we can solve for ππ. This means that two times ππ 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 ππ is 67.45 centimeters.