Triangle Proportionality Theorem

Triangle Proportionality Theorem
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  • 0:05 The Triangle…
  • 0:48 Applying the Theorem
  • 1:39 Drawing a Parallel Line
  • 2:26 Proportional Sides
  • 3:50 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn all about the triangle proportionality theorem and how you can use this interesting theorem to help you solve problems. Learn how a parallel line can create sides that are proportional to each other.

The Triangle Proportionality Theorem

Defined, the triangle proportionality theorem states that if you draw a line parallel to one of the sides of a triangle that also crosses the other two sides, then that line will divide the two sides in a proportional way. If the two sides have been split proportionally, then the ratios of the splits will be equal for both sides.

We can write this mathematically like this: for a triangle ABC with line DE parallel to line BC, the triangle proportionality theorem states that AD / DB = AE / EC.

triangle proportionality theorem

Applying the Theorem

You might be thinking how this applies to you right now. Let me give you a scenario where you can actually use this theorem in real life.

Imagine that you are an engineer, and you have been hired to help finish building a road on a mountain. You are given a drawing of the mountain that shows where roads have already been built. It also shows the distances of each segment of road. There is one segment left to build, and your job is to figure out the distance of that last segment. The mountain in question is located in a distant country, and unfortunately, you can't go and physically measure the distance needed. What can you do? Well, you are going to use the triangle proportionality theorem to help you with your job.

triangle proportionality theorem

Drawing a Parallel Line

The first thing you do is to find a place to draw your parallel line so that it splits two of the triangle's sides proportionally. In this case, your triangle is your mountain. You see that if you draw a line connecting the two villages, Woo and Roo, together, that it will give you a line parallel to the bottom side of the mountain, your triangle. So, you go ahead and draw a line there to help you with your calculations.

triangle proportionality theorem

This job is looking a lot easier to finish now. You have your parallel line, and you see your triangle. That parallel line tells you that the ratios of the splits on both the left and right sides of the triangle are equal.

Proportional Sides

Yes, your side splits are proportional to each other. You use this knowledge to help you with your final calculation.

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