AMSCO'S Geometry. New York - Rye High School

Arcs and Chords 54313-2 ARCS AND CHORDSDEFINITIONA chord of a circle is a line segment whose endpoints are points of the circle.A diameter of a circle is a chord that has the center of the circle as one of itspoints.In the diagram, AB and AOC are chords of circle O.Since O is a point of AOC, AOC is a diameter. Since OAand OC are the lengths of the radius of circle O, OA OCand O is the midpoint of AOC.If the length of the radius of circle O is r, and thelength of the diameter is d, thenCOBAd 5 AOC5 OA 1 OC5 r 1 r5 2rThat is:d 2rThe endpoints of a chord are points on a circle and,therefore, determine two arcs of a circle, a minor arc anda major arc. In the diagram, chord AB, central AOB,minor ABX , and major ABX are all determined by points Aand B. We proved in the previous section that in a circle,congruent central angles intercept congruent arcs. Nowwe can prove that in a circle, congruent central angleshave congruent chords and that congruent arcs havecongruent chords.AOBTheorem 13.3aIn a circle or in congruent circles, congruent central angles have congruentchords.Given(O > (Or andCOD AOB AOBABABProveCD > AB > ArBrCOOD