Chord Theorems of Circles in Geometry

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: How to Find the Measure of an Inscribed Angle

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:01 Chords
  • 0:40 Chord Perpendicular to Radius
  • 1:53 Equidistant Chords
  • 2:38 Examples
  • 3:58 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Speed Speed

Recommended Lessons and Courses for You

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 how a radius will always bisect a perpendicular chord and how equidistant chords will always be congruent to each other. Also, see an example of how you can use these two theorems.


In this lesson, we take a look at two theorems involving chords in circles. We define a chord as a line segment that connects two points on the circle's circumference. Here is what a chord looks like.


A chord can be located anywhere in the circle. In fact, the diameter of a circle is a special chord that passes through the center of the circle.

The two theorems that we will be look at today are these:

  1. If a radius of a circle is perpendicular to a chord in the circle, then the radius bisects the chord.
  2. Two chords are congruent if, and only if, they are equidistant from the center of the circle.

Let's take a closer look.

Chord Perpendicular to Radius

The first theorem says that if a radius of a circle is perpendicular to a chord in the circle, then the radius bisects the chord.


The proof of this theorem relies on the forming of two congruent triangles. First, you know that when you have two perpendicular lines, you will have four right angles. All these angles are congruent to each other. You focus your attention on two of these right angles, specifically angle ADC and angle BDC. You say that these two angles are congruent. Then you say that line segments AC and BC are radii of the circle and therefore, are congruent. Line segment DC is congruent to itself. Now you have two triangles (triangle ADC and triangle BDC), where you know that two of the sides are congruent. Because you know that two of the sides are congruent, it means that the triangles are congruent, and therefore the third side is congruent. This third side happens to be the chord. You say that line segments AD and DB are congruent. This shows that your radius bisects the chord AB because AD is equal to DB making point D the midpoint of chord AB.

Equidistant Chords

This next theorem states that two chords are congruent if, and only if, they are equidistant from the center of the circle.


To prove this theorem, you need to take your two equidistant chords and draw two perpendicular radii, one radius through each chord (radius CE and radius CH). Now you are going to draw your triangles like you did with the previous theorem. You now have four right triangles. Now, because each of your triangles has two congruent sides (sides CF and CD and all the radii), you can show that the third side is also congruent to each other. Because this third side is congruent, you can now show that the chords GI and AB are congruent. And there you have that two equidistant chords are congruent to each other.

To unlock this lesson you must be a Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account