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Lesson 6.2 Chords of a Circle with Arc review

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Lesson 6.2 Chords of a Circle with Arc review Powered By Docstoc
					Geometry


           6.2 Arcs and Chords
           Objectives/Assignment
           • Use properties of arcs of circles..
Geometry


           • Use properties of chords of circles.
           • HOMEWORK Assignment: WS 6.2
            Using Arcs of Circles
           • In a plane, an angle
                                                 central angle
             whose vertex is the
Geometry


             center of a circle is                          A

             a central angle of
             the circle. If the                                      minor
                                                                     arc
             measure of a            major        P
                                     arc
             central angle, APB
             is less than 180°,                                  B
                                             C
             then A and B and the
             points of P
            Using Arcs of Circles
           • in the interior of APB
                                                   central angle
Geometry

             form a minor arc of the
                                                              A
             circle. The points A and
             B and the points of     P
             in the exterior of APB                                   minor
                                                                       arc
                                       major        P
             form a major arc of the arc
             circle. If the endpoints                              B
             of an arc are the                 C
             endpoints of a diameter,
             then the arc is a
             semicircle.
           Naming Arcs                                 G


           • Arcs are named by                             60°
Geometry

             their endpoints. For


                             
             example, the minor     E           60°
                                            H                F
             arc associated with
             APB above is AB.
             Major arcs and
             semicircles are
                                        E
             named by their                     180°

             endpoints and by a
             point on the arc.
           Naming Arcs                                   G


           • For example, the                                60°
Geometry



                
             major arc
                                                  60°


              
             associated with          E
                                              H                F
             APB is A CB .
             EGF here on the
             right is a semicircle.
           The measure of a               E
                                                  180°
             minor arc is defined
             to be the measure of
             its central angle.
           Naming Arcs
                              
           • For instance, m GF=
                                                         G

                                                             60°



               
Geometry

             mGHF = 60°.
           • m GF is read “the        E
                                                  60°
                                              H                    F
             measure of arc GF.”
             You can write the
             measure of an arc next
             to the arc. The              E
                                                  180°
             measure of a
             semicircle is always
             180°.
           Naming Arcs
                             
           • The measure of a GF
                                                       G

                                                           60°
Geometry

             major arc is defined as
             the difference between E           60°




                                 
                                            H                    F
             360° and the measure
             of its associated minor
             arc. For example, m GEF
             = 360° - 60° = 300°.       E
             The measure of the                 180°

             whole circle is 360°.
           Ex. 1: Finding Measures of Arcs

           •   Find the measure

               
Geometry

               of each arc of R.

               
           a. M N

               
           b. MPN
           c. PMN                  N   80°           P
                                                 R




                                             M
           Ex. 1: Finding Measures of Arcs

           •   Find the measure

               
Geometry

               of each arc of R.

               
           a. M N

               
           b. MPN
           c. PMN                   N   80°           P
                                                  R


            
           Solution:
           M N is a minor arc, so
              m M N = mMRN
              = 80°                           M
           Ex. 1: Finding Measures of Arcs

           •   Find the measure

               
Geometry

               of each arc of R.

               
           a. M N

               
           b. MPN
           c. PMN                   N   80°           P
                                                  R


            
           Solution:
           MPN is a major arc, so
              m MPN = 360° – 80°
              = 280°                          M
           Ex. 1: Finding Measures of Arcs

           •   Find the measure

               
Geometry

               of each arc of R.

               
           a. M N

               
           b. MPN
           c. PMN                     N   80°           P
                                                    R

            
           Solution:
           P MN is a semicircle, so
               m PMN= 180°
                                                M
           Note:                                 C


                                           A

           • Two arcs of the same
Geometry

             circle are adjacent if
             they intersect at exactly
             one point. You can add
             the measures of
             adjacent areas.
           • Arc Addition Conjecture


                                                     
                                                         B
             The measure of an arc
             formed by two adjacent
             arcs is the sum of the
             measures of the two
                                         m A BC = m AB+ m BC
             arcs.
           Ex. 2: Finding Measures of Arcs
                                                G
           •   Find the measure of

               
Geometry



               
               each arc.                              H


           a.

                 
              GE                              40°
           b. GEF

               
                                                80°
                                          R
           c. GF                         110°

           m GE = m GH+ m HE =
                                     F                    E
           40° + 80° = 120°
           Ex. 2: Finding Measures of Arcs
                                                    G
           •   Find the measure of

               
Geometry



               
               each arc.                                  H


           a.

                 
              GE                                  40°




               
           b. GEF                                   80°
                                              R
           c. GF                             110°

           m GEF = m GE + m EF       =
                                         F                    E
           120° + 110° = 230°
           Ex. 2: Finding Measures of Arcs
                                                G
           •   Find the measure of

               
Geometry



               
               each arc.                              H


           a.

               
              GE                              40°




                              
           b. GEF

               
                                                80°
                                          R
           c. GF                         110°

           m GF = 360° - m GEF =
                                     F                    E
           360° - 230° = 130°
           Ex. 3: Identifying Congruent Arcs
                                      A

           • Find the measures
Geometry

             of the blue arcs.                        D

             Are the arcs

                
                                          45°
                                  B
             congruent?                         45°


           •

                
               AB and DC are in                       C


            the same circle and

                  
           m AB = m DC= 45°.
            So, AB  DC
           Ex. 3: Identifying Congruent Arcs

           • Find the measures
Geometry

             of the blue arcs.
             Are the arcs

                
                                         80°
             congruent?                        Q
                                     P




                
           •   P Q and RS are in
             congruent circles and

                  
           m P Q = m RS = 80°.
             So, P Q  RS                          R
                                                       80°
                                                             S
            Ex. 3: Identifying Congruent Arcs
            • Find the measures of
              the blue arcs. Are the
Geometry

              arcs congruent?
                                                        Z


              
                                                X




           
           • m XY = m ZW= 65°, but
                                          65°




                  
           XYand ZW are not arcs of the    Y

             same circle or of
              
             congruent circles, so XY
             and ZW are NOT
                                                    W




             congruent.
           Using Chords of Circles
           • A point Y is called
                
Geometry

             the midpoint of
             if XY  YZ . Any


                     
             line, segment, or ray
             that contains Y
             bisects XYZ .
           Chord Arcs Conjecture
           • In the same circle, or in       A
Geometry

             congruent circles, two
             minor arcs are
             congruent if and only if
             their corresponding
             chords are congruent.



            
            AB  BC if and only if       B
                                                 C




            AB  BC
           Perpendicular Bisector of a
           Chord Conjecture
           • If a diameter of a
Geometry

             circle is
             perpendicular to a              F
             chord, then the
             diameter bisects the
             chord and its arc.
                                         E




                  
                                             G
                 DE  EF ,
                 DG  GF            D
           Perpendicular Bisector to a
           Chord Conjecture
           • If one chord is a
Geometry

             perpendicular
                                      J
             bisector of another                M

             chord, then the first
             chord passes
             through the center
             of the circle and is a             K
             diameter.
                      JK is a diameter of   L

                            the circle.
           Ex. 4: Using Chord Arcs Conj.
                                                   (x + 40)°
           • You can use
                  
Geometry

                                   2x°
             Theorem 10.4 to                              C

             find m AD .

                
                                   A
                                              B

           • Because AD  DC,
                
             and AD  DC . So,
             m AD = m DC

              2x = x + 40   Substitute
               x = 40       Subtract x from each
                              side.
           Ex. 5: Finding the Center of a
           Circle
           • Theorem 10.6 can
Geometry

             be used to locate a
             circle’s center as
             shown in the next
             few slides.
           • Step 1: Draw any
             two chords that are
             not parallel to each
             other.
           Ex. 5: Finding the Center of a
           Circle
           • Step 2: Draw the
Geometry

             perpendicular
             bisector of each
             chord. These are
             the diameters.
           Ex. 5: Finding the Center of a
           Circle
           • Step 3: The
Geometry

             perpendicular
             bisectors intersect
             at the circle’s       center


             center.
           Ex. 6: Using Properties of Chords

           • Masonry Hammer. A
Geometry

             masonry hammer has
             a hammer on one end
             and a curved pick on
             the other. The pick
             works best if you
             swing it along a
             circular curve that
             matches the shape of
             the pick. Find the
             center of the circular
             swing.
           Ex. 6: Using Properties of Chords

           • Draw a segment AB,
Geometry

             from the top of the
             masonry hammer to
             the end of the pick.
             Find the midpoint C,
             and draw
             perpendicular bisector
             CD. Find the
             intersection of CD with
             the line formed by the
             handle. So, the center
             of the swing lies at E.
           Chord Distance to the Center
           Conjecture
                                    C
           • In the same circle,                G
Geometry

             or in congruent                        D

             circles, two chords
             are congruent if and       E

             only if they are
             equidistant from the                   B
             center.                        F
                                    A
           • AB  CD if and only
             if EF  EG.
           Ex. 7:
           AB = 8; DE = 8, and   A
Geometry

            CD = 5. Find CF.                     8 F


                                                               B


                                             C



                                                               E
                                         5
                                                           8
                                                       G


                                     D
           Ex. 7:
           Because AB and DE        A
Geometry

             are congruent                          8 F

             chords, they are                                     B
             equidistant from the
             center. So CF                     C


             CG. To find CG,                                      E
             first find DG.                 5
                                                              8
           CG  DE, so CG                                 G


             bisects DE.                D

             Because DE = 8,
                      8
             DG = 2 =4.
           Ex. 7:
           Then use DG to find      A
Geometry

             CG. DG = 4 and                         8 F


             CD = 5, so ∆CGD is                                   B

             a 3-4-5 right                      C

             triangle. So CG = 3.
             Finally, use CG to             5
                                                                  E


             find CF. Because                             G
                                                              8


             CF  CG, CF = CG           D

             =3
           Reminders:
           • Quiz on Friday on 6.1- 6.2
Geometry


           • Yin Yang Due Friday

				
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