AMSCO'S Geometry. New York - Rye High School

546 Geometry of the CircleChords Equidistant from the Center of a CircleWe defined the distance from a point to a line as the length of the perpendicularfrom the point to the line.The perpendicular is the shortest line segment thatcan be drawn from a point to a line. These facts can be used to prove the followingtheorem.Theorem 13.5A diameter perpendicular to a chord bisects the chord and its arcs.Given Diameter COD of circle O, chord AB, and AB ' CD at E.Prove AE > BE, ACX > BC X, and ADX > BD X.ACEOBProof Statements Reasons1. Draw OA and OB. 1. Two points determine a line.2. AB ' CD2. Given.3. AEO and BEO are right 3. Perpendicular lines intersect toangles.form right angles.4. OA > OB4. Radii of a circle are congruent.5. OE > OE5. Reflexive property of congruence.6. AOE BOE 6. HL.7. AE > BE7. Corresponding parts of congruenttriangles are congruent.8. AOE BOE 8. Corresponding parts of congruenttriangles are congruent.9. ACX > BC X9. In a circle, congruent centralangles have congruent arcs.10. AOD is the supplement of 10. If two angles form a linear pair,AOE.then they are supplementary.BOD is the supplement ofBOE.11. AOD BOD 11. Supplements of congruent anglesare congruent.12. ADX > BD X12. In a circle, congruent centralangles have congruent arcs.D