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# Chords in Circles

## Line segments whose endpoints are on a circle.

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Chords in Circles

### Chord Theorems

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, \begin{align*}\overline{BE} \cong \overline{CD}\end{align*} and \begin{align*}\widehat{BE} \cong \widehat{CD}\end{align*}.

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

If \begin{align*}\overline{AD} \perp \overline{BC}\end{align*} and \begin{align*}\overline{BD} \cong \overline{DC}\end{align*} then \begin{align*}\overline{EF}\end{align*} 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 \begin{align*}\overline{EF} \perp \overline{BC}\end{align*}, then \begin{align*}\overline{BD} \cong \overline{DC}\end{align*} and \begin{align*}\widehat{BE} \cong \widehat{EC}\end{align*}.

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 \begin{align*}FE = EG\end{align*} and \begin{align*}\overline{EF} \perp \overline{EG}\end{align*}, then \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{CD}\end{align*} are equidistant to the center and \begin{align*}\overline{AB} \cong \overline{CD}\end{align*}.

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

### Examples

#### Example 1

Find the value of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

The diameter is perpendicular to the chord, which means it bisects the chord and the arc. Set up equations for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

\begin{align*}(3x-4)^\circ& =(5x-18)^\circ \qquad y+4=2y+1\\ 14& =2x \qquad \qquad \qquad \ \ \ 3=y\\ 7& =x \qquad \end{align*}

#### Example 2

\begin{align*}BD = 12\end{align*} and \begin{align*}AC = 3\end{align*} in \begin{align*}\bigodot A\end{align*}. Find the radius.

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

\begin{align*}3^2+6^2& =AB^2\\ 9+36&=AB^2\\ AB&=\sqrt{45}=3\sqrt{5}\end{align*}

#### Example 3

Use \begin{align*}\bigodot A\end{align*} to answer the following.

1. If \begin{align*}m \widehat{BD} = 125^\circ\end{align*}, find \begin{align*}m \widehat{CD}\end{align*}.

\begin{align*}BD = CD\end{align*}, which means the arcs are congruent too. \begin{align*}m \widehat{CD} = 125^\circ\end{align*}.

1. If \begin{align*}m \widehat{BC} = 80^\circ\end{align*}, find \begin{align*}m \widehat{CD}\end{align*}.

\begin{align*}m \widehat{CD} \cong m \widehat{BD}\end{align*} because \begin{align*}BD = CD\end{align*}.

\begin{align*}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{align*}

#### Example 4

Find the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

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

#### Example 5

Find the value of \begin{align*}x\end{align*}.

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

\begin{align*}6x-7& = 35\\ 6x& = 42\\ x& =7\end{align*}

### Review

Fill in the blanks.

1. \begin{align*}\underline{\;\;\;\;\;\;\;\;\;} \cong \overline{DF}\end{align*}
2. \begin{align*}\widehat{AC} \cong \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}\widehat{DJ} \cong \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}\underline{\;\;\;\;\;\;\;\;\;} \cong \overline{EJ}\end{align*}
5. \begin{align*}\angle AGH \cong \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}\angle DGF \cong \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
7. List all the congruent radii in \begin{align*}\bigodot G\end{align*}.

Find the value of the indicated arc in \begin{align*}\bigodot A\end{align*}.

1. \begin{align*}m \widehat{BC}\end{align*}
2. \begin{align*}m\widehat{BD}\end{align*}
3. \begin{align*}m\widehat{BC}\end{align*}
4. \begin{align*}m\widehat{BD}\end{align*}
5. \begin{align*}m\widehat{BD}\end{align*}
6. \begin{align*}m\widehat{BD}\end{align*}

Find the value of \begin{align*}x\end{align*} and/or \begin{align*}y\end{align*}.

1. \begin{align*}AB = 32\end{align*}
2. \begin{align*}AB = 20 \end{align*}
3. Find \begin{align*}m\widehat{AB}\end{align*} in Question 17. Round your answer to the nearest tenth of a degree.
4. Find \begin{align*}m\widehat{AB}\end{align*} 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 \begin{align*}A\end{align*} is the center of the circle.

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Color Highlighted Text Notes

### Vocabulary Language: English Spanish

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.