Identifying Proportional Line Segments

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions, such as Grand Rapids Community College, Pikes Peak Community College, and Austin Peay State University.

We can identify proportional line segments by verifying that their lengths satisfy certain properties and by using helpful corollaries and theorems. This lesson will discuss each of these ways of identifying proportional line segments.

Proportional Line Segments

It's a beautiful day outside, so you and a friend decide to find a park to go for a walk in! You hop in your car and set out to find a nice park. As you're driving along, you see a sign for Proportional Park. You're glad to have found a park to walk in, but you're curious about its name. At the trail head of the park, there is a map of the different walking trails along with a description of the park itself.


Ah-ha! That's where its name came from! The trails form a triangle with a trail connecting two of the sides of the triangle that is parallel to the third side in such a way that the line segments it creates are proportional.

Proportional line segments happen when we have four line segments, call them a, b, c, and d, such that the following is true:

a/b = c/d

We call a/b and c/d ratios, and the equation indicating the two ratios are equal is called a proportion.

Take a look at the park map again. We have the following lengths for the trails:

  • Trail A = 1 mile
  • Trail B = 3 miles
  • Trail C = 2 miles
  • Trail D = 6 miles

Now consider the ratios comparing trail A to trail B and comparing trail C to trail D.

  • A/B = 1/3
  • C/D = 2/6 = 1/3

We see that the two ratios are equal, so we have the proportion A/B = C/D, and the line segments A, B, C, and D are proportional. Now that you know why the park has its name, you have to admit that it's a pretty clever name!

When we're given lengths of line segments as we just were, we can identify proportional line segments by verifying that the ratios of their lengths are equal. However, if we aren't given lengths, we can still identify proportional line segments in various scenarios. Let's take a look at a couple of rules that help us to do just that!

Triangles and Proportional Line Segments

As it turns out we have a rule that would allow us to know that trails A, B, C, and D are proportional without knowing their lengths. There is a corollary, let's call it the triangle proportional line segment corollary, that states that if a line segment is drawn connecting two sides of a triangle in such a way that it is parallel to the third side, then it divides the two connected sides into proportional line segments.


As we saw, the park map shows that trail F connects two sides of the triangle that the trails form, and it is parallel to trail E, which is the third side of the triangle, so the triangle proportional line segment corollary tells us that trail F divides the two trails that it connects into proportional line segments, so we would know that A/B = C/D even if we didn't have numbers to verify this. Pretty handy, huh?

Let's take a look at another rule that helps us to identify proportional line segments.

The Proportional Segments Theorem

Another rule that helps us to identify proportional line segments is the proportional segments theorem. This theorem states that when we cut (intersect) two transversals (lines) with three or more parallel lines we divide the transversals into proportional line segments.


We can use this theorem to identify proportional lines segments and use it in various applications. For example, suppose there is a Proportional Park II, and it's trails consist of two transversals cut by three parallel lines as shown in the image.


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