Chord Theorems of Circles in Geometry

Chord Theorems of Circles in Geometry
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  • 0:01 Chords
  • 0:40 Chord Perpendicular to Radius
  • 1:53 Equidistant Chords
  • 2:38 Examples
  • 3:58 Lesson Summary
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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.

Chords

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.

chord

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.

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.

chord

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.

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