Segment Lengths in Circles

Instructor: Yuanxin (Amy) Yang Alcocer

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

When it comes to circles and segments, three rules help you to figure out the lengths of these segments. Learn how interesting the relationship between the segments is in this lesson.

Segments in Circles

In this lesson, you'll learn about the relationships that segments in circles have with each other. By definition, a segment is a part of a line. There are several different types of segments that you can have when it comes to circles. Here is a picture showing them.

circle segments

The green number 1 segment is called a chord. Its endpoints are both on the edge of the circle. The orange number 2 segment is called a secant. It's basically an extended chord. The pink number 3 segment is called a tangent. It is a segment that touches the edge of the circle.

Three different combinations of these segments create interesting relationships that you'll learn about in just a moment.

Intersecting Chords

The first is that of the intersecting chords. When you have two chords that intersect each other inside a circle, the relationship the parts of each segment have will always be this:

  • The product of the parts of one chord is equal to the product of the parts of the other chord.

Here is a picture showing how two intersecting chords look in a circle.

circle segments

Writing out the relationship algebraically, you get this:

  • a * b = c * d

You see how each chord now has two parts because each chord has been intersected by the other. This relationship says that if you multiply the two parts of each chord, they will always be equal to each other.

You can use this information to help you find missing lengths. For example, say you are given the lengths of a, b, and c. You need to find the length of d. Well, you can use this relationship and plug in your values for a, b, and c and then use algebra to solve for d.

Let's take a look.

You are given this:

  • a = 3, b = 5, c = 4

To find d, you plug in your a, b, and c values into your relationship and solve for d. Like this:

  • 3 * 5 = 4 * d
  • 15 = 4d
  • d = 15 / 4

And, you are done.

Two Secants

The second interesting relationship is when you have two secants that intersect each other outside the circle. When this happens, you get this relationship:

  • The exterior portion of the first secant times the entire first secant is equal to the exterior portion of the second secant times the entire second secant.

Drawing it out, it looks like this:

circle segments

Algebraically, the relationship looks like this:

  • a * b = c * d

Yes, the algebraic relationship looks just like the one when you have two intersecting chords. If you think about it, it makes sense since your secants are basically extended chords.

You use this relationship the same way you use the relationship for your intersecting chords.

For example, say you are given b, c, and d. You can then use this relationship to find a.

If you are given this:

  • b = 10, c = 3, d = 8

Your a is then equal to this:

  • a * 10 = 3 * 8
  • 10a = 24
  • a = 24 / 10
  • a = 12 / 5

And, you are done.

Secant and Tangent

The third interesting relationship is when you have a secant and a tangent that intersect outside the circle. When this happens, you have this relationship:

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