Segment Relationships 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.

After reading this lesson, you'll understand how useful segments and circles are. You'll learn that when you have segments in a circle, you get some very interesting relationships that help you figure out various values.

Segments in Circles

In this lesson, you'll learn about the relationships that form when you combine segments and circles together. A segment is a line that has a beginning and an end. A circle is a flat round shape. There are two types of segments you can have that cross your circle. A secant segment is a segment that intersects the edge of your circle twice. A tangent segment is a segment that touches the edge of your circle once. A tangent segment never passes through a circle.

There are actually three different relationships your segments and circles can have. These three different relationships are actually theorems as they are proven relationships that work all the time. Let's take a look at these three.

Intersecting Segments

The first scenario is when you have two secant segments that intersect each other inside the circle.


segment relationships


When you have intersecting segments such as these, the relationship is that the product of the segment pieces of one segment is equal to the product of the segment pieces of the other.

  • (segment piece a) * (segment piece b) = (segment piece c) * (segment piece d)
  • a * b = c * d

You can use this relationship to help you solve problems that ask you find the length of a missing segment piece. For example, if you are given values for a, b, and d, but not c, you'll be asked to use this relationship to help you figure out the length of segment piece c.


segment relationships


To answer this problem, you plug in your values for a, b, and d and then use algebra to solve for your variable c. Looking at the above picture, you see that your a value is 4, your b value is 6, and your d value is 8. You plug these into your relationship and then you can solve for your c value.

  • a * b = c * d
  • 4 * 6 = c * 8
  • 24 = 8c
  • c = 3

And you are done!

Two Secants

The next scenario is when you have two secant segments that intersect outside the circle.


segment relationships


The a and the c are the segment pieces that are outside the circle. The b and d are the whole segment including the a and c segment pieces. So, b includes a, and d includes c. The relationship here is that the product of a whole secant segment with its external part is equal to the product of the other whole secant segment with its external part.

  • a * b = c * d

If you compare this to the first relationship, they are very similar. You use this relationship just like you would use the first.

For example, say you are given this information:


segment relationships


You can use this second relationship to help you find the missing value d. Your a is 5 and your b is 8. Your c is 4. So, plugging these in and solving for d, you get this:

  • a * b = c * d
  • 5 * 8 = 4 * d
  • 40 = 4d
  • d = 10

Your d is 10.

One Secant and One Tangent

The third relationship is when you have one secant segment and one tangent segment.


segment relationships


The relationship here is that the product of the whole secant segment with its external part is equal to the square of the tangent segment.

  • b * c = a2

Just like with the other relationships, this is best used to find missing values.

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