Page 162 - Math Course 3 (Book 2)
P. 162

Circles: Arcs and Chords





         MO. 11 - L3b                                       Since radius WL is perpendicular to chord HK,
                                                            HL ≅ LK.
            The Relationships Between

                Chords and Diameters                           mMK + mKL = mMKL    Arc addition postulate

                                                            mMK + mMHL = mMKL      Substitution
         THEOREM 11.3

                                                                    mMK + 53 = 180  Substitution
         In a circle, if a diameter (or radius) is perpendicular
         to a chord, then it bisects the chord and its arc.                  mMK = 127  Subtract 53
                                                                                   from each side.
         Example:
                                                              Answer             mMK ≈ 127
         If BA ⟂ TV, then UT ≅ UV and AT ≅ AV.
                             B
                                                           Circle W has a radius of 10 centimeter, Radius WL is
                                                           is perpendicular to chord HK, which is 16 centime-
                                                           ters long. Find JL.
                               C                                               M

                               U
                 T                       V


                             A                                                   W
                                                                                  J
                                                                     H                     K
         THEOREM 11.4                                                           L


         In a circle or in congruent circles, two chords are    Draw radius WK. △WJK is a right triangle.
         congruent if and only if they are equidistant from
         the center.
                                                            WK = 10           r = 10

                      Let’s Begin                           ML bisects HK     A radius perpendicular to a
                                                                              chord bisects it.

                                                                 1
                                                            JK =       (HK)   Definition of segment bisector
                                                                 2
        Radius Perpendicular to a Chord                          1
                                                                  =       (16) or 8  HK = 16
                                                                 2
            Example
                                                            Use the Pythagorean Theorem to find WJ.
         Circle W has a radius of 10 centimeter, Radius WL
         is perpendicular to chord HK, which is 16 centime-  (WJ)² + (JK)² = (WK)²  Pythagorean Theorem
         ters long. If mHL ≈ 53, find mMK.                       (WJ)² + 8² = 10²  JK = 8, WK = 10
                               M
                                                                (WJ)² + 64 = 100  Simplify.
                                                                                 Subtract 64 from each
                                                                         (WJ)² = 36
                                 W                                               side.
                                 J
                      H                  K                                    WJ = 6  Take the square root of
                                                                                 each side.
                                L
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