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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.

    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.

    f-d_19e5e04c404c16527fd5c5d56ead6a6d4920136a9b025b6e667e4445+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    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 \(\PageIndex{7}\)
    1. If \(m\widehat{BD}=125^{\circ}\), find \(m\widehat{CD}\).
    2. If \(m\widehat{BC}=80^{\circ}\), find \(m\widehat{CD}\).

    Solution

    1. \(BD=CD\), which means the arcs are congruent too. \(m\widehat{CD}=125^{\circ}\).
    2. \(m\widehat{CD}\cong m\widehat{BD}\) 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 \(\PageIndex{10}\)
    1. \(\text{_____}\cong \overline{DF}\)
    2. \(\widehat{AC} \cong \text{_____}\)
    3. \(\widehat{DJ}\cong \text{_____}\)
    4. \(\text{_____}\cong \overline{EJ}\)
    5. \(\angle AGH\cong \text{_____}\)
    6. \(\angle DGF\cong \text{_____}\)
    7. List all the congruent radii in \(\bigodot G\).

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

    1. \(m\widehat{BC}\)
      f-d_c0e07b24726f78db00fccf22e1f3918180a27ab379bd20cede393394+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{11}\)
    2. \(m\widehat{BD}\)
      f-d_bce947699ed17e4af5984ac250939bce56a798366107e63de77029c5+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{12}\)
    3. \(m\widehat{BC}\)
      f-d_c1cd036cd92993bb5ac74ba6504eaaf82ff93aa82c9c011ee8938f8c+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{13}\)
    4. \(m\widehat{BD}\)
      f-d_22fe07967bfa0b508d63993b168b76d2316f2cdd0d5144e66a1f98b0+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{14}\)
    5. \(m\widehat{BD} \)
      f-d_3ed1e002c73425868c9e3abb33a7391ef4a477bde1a3e1ca0dbf105e+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{15}\)
    6. \(m\widehat{BD}\)
      f-d_4f8a37ea252641518cae7abf9e4885e448590d80a944ca8d69932b23+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{16}\)

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

    1. f-d_95516e6a2de73e7e3446d679a0d9b1f3111de133bc0cf62b5a7fc83b+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{17}\)
    2. f-d_d47ecdb188af6f2ba1d054cd698e394d84eb04b4f4638b6d0767b4e2+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{18}\)
    3. f-d_9be9abda5a3dbc885f1f2cd569efbbf5fc8ac77202c1b9a5f088667b+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{19}\)
    4. \(AB=32\)
      f-d_ea7a725b547ada9e571cd08f2aa773e4d8755e54460b6a818cfc727c+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{20}\)
    5. f-d_4492ff4d528ec529326136c74f79977f45dd9b253385402237da0e9c+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{21}\)
    6. f-d_8f018a04724318888c2af6dab8e56abf73852ca4f411bf60d1f003fc+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{22}\)
    7. f-d_56aee6101f23de087cdb1cf8e4c4128722b26f889fa0706c5874e346+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{23}\)
    8. f-d_737eb20a8a8d654e96f6dc7cc4bc884a737733cb0201d3d4f039bcc9+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{24}\)
    9. \(AB=20\)
      f-d_158517bba0cbbcd29603245c2ac279bee074d72c72505a8b21644cf3+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{25}\)
    10. Find \(m\widehat{AB}\) in Question 17. Round your answer to the nearest tenth of a degree.
    11. 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.

    1. f-d_eb4b17b2ae37553bd45a56c987091014a1018bd3481fa03047b60e08+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{26}\)
    2. f-d_f73f80551beeb5404eb70ce01f9b8b70fba384ebd8c21685d435c671+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{27}\)
    3. f-d_cd466baf5e5be089161117b67b6f950de91de043fb4b82a3506276a9+IMAGE_TINY+IMAGE_TINY.png
      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