Arcs and Chords

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					  Lesson 8-4

   Arcs
and Chords
   Lesson 8-4: Arcs and Chords   1
   Theorem #1:
In a circle, if two chords are congruent then their corresponding
minor arcs are congruent.
                                                       A           B
    If AB  CD then AB  CD                                E

                                                       C
                                                                   D
Example:   Given mAB  127 , find the mCD.

             Since mAB  mCD
             mCD  127
                         Lesson 8-4: Arcs and Chords           2
    Theorem #2:
 In a circle, if a diameter (or radius) is perpendicular to a
 chord, then it bisects the chord and its arc.
    If DC  AB then DC bi sec ts AB and AB.
     AE  BE and AC  BC                                    D


Example: If AB = 5 cm, find AE.                          A       E       B

          If mAB  120 , find mAC.                           C
                                    AB       5
                           AE          AE   2.5 cm
                                     2       2
                                  m AB            120
                           m AC       ,  m AC       60
                                    2              2
                           Lesson 8-4: Arcs and Chords               3
  Theorem #3:
In a circle, two chords are congruent if and only if
they are equidistant from the center.            D
                                                        F

    CD  AB iff OF  OE
                                                C
                                                        O


                                                    A       E       B
Example:   If AB = 5 cm, find CD.

       Since AB = CD, CD = 5 cm.


                      Lesson 8-4: Arcs and Chords               4
    Try Some Sketches:
 Draw a circle with a chord that is 15 inches long and 8 inches from
  the center of the circle.
 Draw a radius so that it forms a right triangle.

 How could you find the length of the radius?

 Solution: ∆ODB is a right triangle and OD bi sec ts AB
          AB 15
     DB=      = =7.5 cm                                  A   15cm
                                                                               B
           2    2                                            D
     OD=8 cm                                                     8cm       x
     OB2 =OD2 +DB2                                           O

     OB2 =82 +(7.5) 2 =64+56.25=120.25
     OB= 120.25  11cm

                           Lesson 8-4: Arcs and Chords                 5
     Try Some Sketches:
 Draw a circle with a diameter that is 20 cm long.
 Draw another chord (parallel to the diameter) that is 14cm long.

 Find the distance from the smaller chord to the center of the circle.
                                      AB 14
Solution: OE bi sec ts AB.  EB            7cm
                                       2    2
 ∆EOB is a right triangle. OB (radius) = 10 cm A         14 cm
                                                                    B
      OB 2  OE 2  EB 2                                  E
                                                          x
                                                               10 cm
       10  X  7
         2     2    2                                                        10 cm
                                                           C      20cm   O           D

       X 2  100  49  51
       X  51  7.1 cm
                             Lesson 8-4: Arcs and Chords                 6

				
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