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

# Question Video: Finding the Unknown Lengths in a Triangle given the Other Sidesβ Lengths Using the Relations of Parallel Lines Mathematics • First Year of Secondary School

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Given that π΄π΅πΆπ· is a parallelogram, find the length of line segment ππ.

04:14

### Video Transcript

Given that π΄π΅πΆπ· is a parallelogram, find the length of line segment ππ.

In this question, we need to find this line segment ππ. You may think that this is going to be quite difficult. After all, weβre 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΅πΆπ· is a parallelogram. So letβ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 π·ππΆ, we arenβt given the information that ππ is parallel to π·πΆ. 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 letβs look at whatβs happening in this triangle π·ππΆ. From the markings on the diagram, we can see that π·π is exactly equal to ππ. And so we could say that π·π is half of the whole length of π·π. In the same way, πΆπ is equal to ππ. So πΆπ is half of the whole length of πΆπ. So the length of ππ has split both of the other two sides into two equal pieces.

As an aside, however, the side splitter theorem doesnβ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 letβs return to this parallelogram. Because we have the two sides ππ· and ππΆ in the larger triangle divided into equal proportions, then we have that the line segment π·πΆ is parallel to the line segment ππ. We can also note that triangle πππ is similar to triangle ππ·πΆ.

Since π΄π΅πΆπ· is a parallelogram, we know that there are two opposite sides of equal lengths. So the length of line segment π·πΆ is also 134.9 centimeters. Now, letβs see if we can work out any unknown lengths and letβs denote this length of ππ 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 letβs include one of the sides of which we know its actual length. And so ππ over ππ· must be equal to the proportionality ππ over π·πΆ.

We defined ππ as π and ππ· as two π. And then on the right-hand side, we donβt yet know ππ, but we do know that π·πΆ is 134.9 centimeters. We have a few unknowns here. But donβt 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 ππ. This means that two times ππ 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 ππ is 67.45 centimeters.

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