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High School Geometry: Homework Help Resource13 chapters | 142 lessons

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.

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.

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.

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.

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 = 4
*d* *d*= 15 / 4

And, you are done.

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:

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- 10
*a*= 24 *a*= 24 / 10*a*= 12 / 5

And, you are done.

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:

- The exterior part of the secant times the entire secant is equal to the square of the tangent.

Drawing it out, it looks like this:

The relationship written out algebraically, is this one:

*a***b*=*c*2

If you are given just two of these values, then you'll be able to find the third value. For example, if you are given this:

*c*= 4 and*a*= 3

Then you can calculate your *b* by plugging in your value for *a* and *c* and then solving for *b* like this:

- 3 *
*b*= 42 - 3
*b*= 16 *b*= 16 / 3

And, you are done.

Let's review.

A **segment** is a part of a line. When you combine segments with circles, you get three different types of segments. You have the **chord**, a segment whose endpoints are the edges of the circle. Then, you have the **secant**, basically an extended chord. And, you have the **tangent**, a segment that touches the edge of the circle.

Here is a table summarizing the three interesting relationships you get when you combine these segments:

Combination | Relationship |
---|---|

Two intersecting chords | The product of the parts of each segment is always equal to each other |

Two secants that intersect outside the circle | The exterior part of one secant times the entire secant is equal to the exterior part of the other secant times the entire secant |

A secant and tangent that intersect outside the circle | The exterior part of the secant times the whole secant is equal to the square of the tangent |

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High School Geometry: Homework Help Resource13 chapters | 142 lessons

- Circles: Area and Circumference 8:21
- Circular Arcs and Circles: Definitions and Examples 4:36
- Central and Inscribed Angles: Definitions and Examples 6:32
- Measure of an Arc: Process & Practice 4:51
- How to Find the Measure of an Inscribed Angle 5:09
- Tangent of a Circle: Definition & Theorems 3:52
- Measurements of Angles Involving Tangents, Chords & Secants 6:59
- Measurements of Lengths Involving Tangents, Chords and Secants 5:44
- Inscribed and Circumscribed Figures: Definition & Construction 6:32
- Arc Length of a Sector: Definition and Area 6:39
- Segment Lengths in Circles
- Go to Circular Arcs and Circles: Homework Help

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