AMSCO'S Geometry. New York - Rye High School

Arcs and Chords 547Since a diameter is a segment of a line, the following corollary is also true:Corollary 13.5aTheorem 13.6A line through the center of a circle that is perpendicular to a chord bisectsthe chord and its arcs.An apothem of a circle is a perpendicular line segmentfrom the center of a circle to the midpoint of a chord. Theterm apothem also refers to the length of the segment. In thediagram, E is the midpoint of chord AB in circle O,AB ' CD, and OE, or OE, is the apothem.The perpendicular bisector of the chord of a circle contains the center of thecircle.ACEDOBGiven Circle O, and chord AB with midpoint M and perpendicularbisector k.AMBProve Point O is a point on k.OProof In the diagram, M is the midpoint of chord AB in circle O.kThen, AM = MB and AO = OB (since these are radii). Points O and M areeach equidistant from the endpoints of AB. Two points that are each equidistantfrom the endpoints of a line segment determine the perpendicular bisectorof the line segment. Therefore, OM g is the perpendicular bisector of AB.Through a point on a line there is only one perpendicular line. Thus, OM g andk are the same line, and O is on k, the perpendicular bisector of AB.Theorem 13.7aIf two chords of a circle are congruent, then they are equidistant from thecenter of the circle.Given Circle O with AB > CD, OE ' AB, and OF ' CD.ProveProofOE > OFF DA line through the center of a circle that is perpendicular to a Cchord bisects the chord and its arcs. Therefore, OE gand OF gbisect the congruentchords AB and CD. Since halves of congruent segments are congruent,EB > FD. Draw OB and OD. Since OB and OD are radii of the same circle,OB > OD. Therefore, OBE ODF by HL, and OE > OF.AEOB