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6: Circles
6.12: Chords and Central Angle Arcs

6.12: Chords and Central Angle Arcs

Last updated

Jun 15, 2022
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- 6.11: Arc Length
- 6.13: Segments from Chords

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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.

f-d_2ea1cf98cd7bab0d6cd479df866ce1d1371c7edd47866c18a95bbfd3+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{1}$

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.

f-d_19e5e04c404c16527fd5c5d56ead6a6d4920136a9b025b6e667e4445+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{2}$

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.

f-d_99e22c456e3f4815417f855a0311378b46e0787fd21b0120f5151699+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{4}$

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 $y$ .

f-d_a040b0389542bb379fa6c030342257c30c5de4c41e22004f5236607e+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{5}$

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.

f-d_e0b218827409e0747e0c8ecebad5c4eab55aa4db23adff4bc47203d4+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{6}$

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.

f-d_ca0baf3d34c480e5bd5781f6dd90473b3da0b492c4bbfe46186ef20e+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{7}$

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 $y$ .

f-d_5322912138f6a893d14b0f2210f643adc820904ff5e0813b4e9f0de8+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{8}$

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 $x$ .

f-d_4c6c48ef93ea52ed1fe355459e5aaaf3616653d3ee5a6959905a9efc+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{9}$

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.

f-d_7fd8fcdc43f0c3daab41774c107f386a3a4cda83d1877c7f19190cd0+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{10}$

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

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

$m\widehat{BC}$

Figure $\PageIndex{11}$
$m\widehat{BD}$

Figure $\PageIndex{12}$
$m\widehat{BC}$

Figure $\PageIndex{13}$
$m\widehat{BD}$

Figure $\PageIndex{14}$
$m\widehat{BD}$

Figure $\PageIndex{15}$
$m\widehat{BD}$

Figure $\PageIndex{16}$

Find the value of $x$ and/or $y$ .

Figure $\PageIndex{17}$
Figure $\PageIndex{18}$
Figure $\PageIndex{19}$
$AB=32$

Figure $\PageIndex{20}$
Figure $\PageIndex{21}$
Figure $\PageIndex{22}$
Figure $\PageIndex{23}$
Figure $\PageIndex{24}$
$AB=20$

Figure $\PageIndex{25}$
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.

Figure $\PageIndex{26}$
Figure $\PageIndex{27}$
Figure $\PageIndex{28}$

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

Chords in Circles

Chord Theorems

Review

Review (Answers)

Vocabulary

Additional Resources

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