# 6.12: Chords and Central Angle Arcs

- Page ID
- 5027

Arcs determined by angles whose vertex is the center of a circle and chords (segments that connect two points on a circle).

## Chords in Circles

### Chord Theorems

There are several important theorems about chords that will help you to analyze circles better.

1. **Chord Theorem #1:** In the same **circle** or congruent circles, minor arcs are congruent if and only if their corresponding chords are congruent.

In both of these pictures, \(\overline{BE}\cong \overline{CD}\) and \(\widehat{BE}\cong \widehat{CD}\).

2. **Chord Theorem #2:** The perpendicular bisector of a chord is also a **diameter**.

If \(\overline{AD}\perp \overline{BC}\) and \(\overline{BD}\cong \overline{DC}\) then \(\overline{EF}\) is a diameter.

3. **Chord Theorem #3:** If a diameter is perpendicular to a chord, then the diameter bisects the chord and its corresponding arc.

If \(\overline{EF}\perp \overline{BC}\), then \(\overline{BD}\cong \overline{DC}\)

4. **Chord Theorem #4:** In the same circle or congruent circles, two chords are congruent if and only if they are equidistant from the center.

The shortest distance from any point to a line is the perpendicular line between them. If \(FE=EG\) and \(\overline{EF}\perp \overline{EG}\), then \(\overline{AB}\) and \(\overline{CD}\) are equidistant to the center and\(\overline{AB}\cong \overline{CD}\).

What if you were given a circle with two chords drawn through it? How could you determine if these two chords were congruent?

Example \(\PageIndex{1}\)

Find the value of \(x\) and \(y\).

**Solution**

The diameter is perpendicular to the chord, which means it bisects the chord and the arc. Set up equations for \(x\) and \(y\).

\(\begin{array}{rlr}

(3 x-4)^{\circ} & =(5 x-18)^{\circ} & y+4=2 y+1 \\

14 & =2 x & 3=y \\

7 & =x

\end{array}\)

Example \(\PageIndex{2}\)

\(BD=12\) and \(AC=3\) in \(\bigodot A\). Find the **radius**.

**Solution**

First find the radius. \(\overline{AB}\) is a radius, so we can use the right triangle \Delta ABC\) with hypotenuse \(\overline{AB}\). From Chord Theorem #3, \(BC=6\).

\(\begin{aligned} 3^2+6^2&=AB^2 \\ 9+36&=AB^2 \\ AB&=\sqrt{45}=3\sqrt{5}\end{aligned}\)

Example \(\PageIndex{3}\)

Use \(\bigodot A\) to answer the following.

- If \(m\widehat{BD}=125^{\circ}\), find \(m\widehat{CD}\).
- If \(m\widehat{BC}=80^{\circ}\), find \(m\widehat{CD}\).

**Solution**

- \(BD=CD\), which means the arcs are congruent too. \(m\widehat{CD}=125^{\circ}\).
- \(m\widehat{CD}\cong m\widehat{BD}\) because \(BD=CD\).

\(\begin{aligned} m\widehat{BC}+m\widehat{CD}+m\widehat{BD}&=360^{\circ} \\ 80^{\circ}+2m\widehat{CD}&=360^{\circ} \\ 2m\widehat{CD}&=280^{\circ} \\ m\widehat{CD}=140^{\circ}\end{aligned}\)

Example \(\PageIndex{4}\)

Find the values of \(x\) and \(y\).

**Solution**

The diameter is perpendicular to the chord. From Chord Theorem #3, \(x=6\) and \(y=75^{\circ}\).

Example \(\PageIndex{5}\)

Find the value of \(x\).

**Solution**

Because the distance from the center to the chords is equal, the chords are congruent.

\(\begin{aligned} 6x−7&=35 \\ 6x&=42 \\ x&=7 \end{aligned}\)

## Review

Fill in the blanks.

- \(\text{_____}\cong \overline{DF}\)
- \(\widehat{AC} \cong \text{_____}\)
- \(\widehat{DJ}\cong \text{_____}\)
- \(\text{_____}\cong \overline{EJ}\)
- \(\angle AGH\cong \text{_____}\)
- \(\angle DGF\cong \text{_____}\)
- List all the congruent radii in \(\bigodot G\).

Find the value of the indicated arc in \(\bigodot A\).

- \(m\widehat{BC}\)
- \(m\widehat{BD}\)
- \(m\widehat{BC}\)
- \(m\widehat{BD}\)
- \(m\widehat{BD} \)
- \(m\widehat{BD}\)

Find the value of \(x\) and/or \(y\).

- \(AB=32\)
- \(AB=20\)
- Find \(m\widehat{AB}\) in Question 17. Round your answer to the nearest tenth of a degree.
- Find \(m\widehat{AB}\) in Question 22. Round your answer to the nearest tenth of a degree.

In problems 25-27, what can you conclude about the picture? State a theorem that justifies your answer. You may assume that A is the center of the circle.

## Review (Answers)

To see the Review answers, open this PDF file and look for section 9.4.

## Vocabulary

Term | Definition |
---|---|

chord |
A line segment whose endpoints are on a circle. |

circle |
The set of all points that are the same distance away from a specific point, called the .center |

diameter |
A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. |

radius |
The distance from the center to the outer rim of a circle. |

## Additional Resources

Interactive Element

Video: Chords in Circles Principles - Basic

Activities: Chords in Circles Discussion Questions

Study Aids: Circles: Segments and Lengths Study Guide

Practice: Chords and Central Angle Arcs