# Question Video: Finding the Lengths of Proportional Line Segments between Parallel Lines Mathematics • 11th Grade

In triangle π΄π΅πΆ, π΄πΈ = 1 cm, πΈπΆ = 2 cm, and π΄π΅ = 6 cm. Find the length of line segment π΄πΉ.

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### Video Transcript

In triangle π΄π΅πΆ, π΄πΈ equals one centimeters, πΈπΆ equals two centimeters, and π΄π΅ equals six centimeters. Find the length of line segment π΄πΉ.

In this diagram involving the larger triangle π΄π΅πΆ, we can also see that there are two parallel line segments, πΈπΉ and πΆπ΅. We are also given the lengths of three different line segments. π΄πΈ is one centimeter, πΈπΆ is two centimeters, and π΄π΅ is six centimeters. So we can add these onto the diagram. We are asked to find the length of the line segment π΄πΉ, which is part of this smaller triangle π΄πΈπΉ. To do this, weβll need to use the parallel lines to help us. These pair of parallel lines will allow us to apply the side splitter theorem.

This theorem tells us that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally. The line which is parallel to one side of the triangle is the line segment πΈπΉ, because itβs parallel to the line segment π΅πΆ. Because it intersects the other two sides, then these other two sides, π΄πΆ and π΄π΅, are divided proportionally by the line segment πΈπΉ. We could therefore write the proportionality statement that π΄πΈ over π΄πΆ is equal to π΄πΉ over π΄π΅. Substituting in the given length measurements will allow us to work out the unknown length of π΄πΉ.

Take care with the substitutions because the line segment π΄πΆ will be the sum of the line segments π΄πΈ of one centimeters and πΈπΆ of two centimeters. We therefore have one-third is equal to π΄πΉ over six. Cross multiplying, we have six is equal to three times π΄πΉ. And dividing both sides by three, we have two equals π΄πΉ. We can then give the answer that the length of the line segment π΄πΉ is two centimeters.

Before we finish with this question, itβs worth noting that there are alternative proportionality statements that we could have written. We could have said, for example, that π΄πΈ over π΄πΉ is equal to πΈπΆ over πΉπ΅. Substituting the values on the left side, π΄πΈ is one centimeter and π΄πΉ is the line segment that we wish to work out. On the right-hand side, we know that πΈπΆ is two centimeters. The length of πΉπ΅ is one that we donβt know, but we can write it in terms of π΄πΉ as six minus π΄πΉ. Writing it like this allows us to have just one unknown in our equation.

Cross multiplying, we have six minus π΄πΉ is equal to two times π΄πΉ. Then we can add π΄πΉ to both sides, which gives us six is equal to two times π΄πΉ plus π΄πΉ. And we know that two π΄πΉ plus π΄πΉ is simply three π΄πΉ. Dividing both sides by three, we have two is equal to π΄πΉ. And so we have demonstrated how two different proportionality statements will allow us to find that the length of the line segment π΄πΉ is two centimeters.