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Chords in Circles

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Chords in Circles

What if you were given a circle with two chords drawn through it? How could you determine if these two chords were congruent? After completing this Concept, you'll be able to use four chord theorems to solve problems like this one.

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Chords in Circles CK-12

Guidance

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} and \widehat{BE} \cong \widehat{EC} .

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} .

Example A

Use \bigodot A to answer the following.

a) If m \widehat{BD} = 125^\circ , find m \widehat{CD} .

b) If m \widehat{BC} = 80^\circ , find m \widehat{CD} .

Answers:

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

b) m \widehat{CD} \cong m \widehat{BD} because BD = CD .

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

Example B

Find the values of x and y .

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

Example C

Find the value of x .

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

6x-7& = 35\\6x& = 42\\x& =7

Chords in Circles CK-12

Guided Practice

1. Find the value of x and y .

2. BD = 12 and AC = 3 in \bigodot A . Find the radius.

3. Find m\widehat{BD} from #2.

Answers:

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

(3x-4)^\circ& =(5x-18)^\circ \qquad y+4=2y+1\\14& =2x \qquad \qquad \qquad \ \ \ 3=y\\7& =x \qquad

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

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

3. First, find the corresponding central angle, \angle BAD . We can find m \angle BAC using the tangent ratio. Then, multiply m\angle BAC by 2 for m\angle BAD and m\widehat{BD} .

\tan^{-1} \left ( \frac{6}{3} \right ) & = m\angle BAC\\m\angle BAC & \approx 63.43^\circ\\m\angle BAD & \approx 2 \cdot 63.43^\circ \approx 126.86^\circ \approx m\widehat{BD}

Practice

Fill in the blanks.

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

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

  1. m \widehat{BC}
  2. m\widehat{BD}
  3. m\widehat{BC}
  4. m\widehat{BD}
  5. m\widehat{BD}
  6. m\widehat{BD}

Find the value of x and/or y .

  1. AB = 32
  2. AB = 20
  3. Find m\widehat{AB} in Question 17. Round your answer to the nearest tenth of a degree.
  4. 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.

Vocabulary

chord

chord

A line segment whose endpoints are on a circle.
circle

circle

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

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

radius

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

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