AMSCO'S Geometry. New York - Rye High School

482 Ratio, Proportion, and SimilarityFDACGEB• Let BC 4y. Then BE 2y, EC 2y, EG y, and GC x.Also, BG 2y y 3y.FCGC• Therefore, and BG 5 y AF 5 3x x 5 1 33y 5 1 3.FC GC1 FCSince AF and BG are each equal to , AF 5 GC3 BG. We say that the points F andG divide AC and BC proportionally because these points separate the segmentsinto parts whose ratios form a proportion.DEFINITIONTwo line segments are divided proportionally when the ratio of the lengths of theparts of one segment is equal to the ratio of the lengths of the parts of the other.The points D and E also divide AC and BC proportionally because thesepoints also separate the segments into parts whose ratios form a proportion.ADDC 5 2x2x 5 1ADTherefore, DC 5 BEEC.andBEEC 5 2y2y 5 1Theorem 12.3aIf two line segments are divided proportionally, then the ratio of the lengthof a part of one segment to the length of the whole is equal to the ratio ofthe corresponding lengths of the other segment.ABGiven ABC and DEF with BC 5 DEEF.ABCProveABAC 5 DEDFDEFProof Statements Reasons1.ABBC 5 DEEF1. Given.2. (AB)(EF) (BC)(DE) 2. The product of the meansequals the product of theextremes.3. (AB)(EF) (BC)(DE) 3. Addition postulate. (AB)(DE) (AB)(DE)4. (AB)(EF DE) (DE)(BC AB) 4. Distributive property.5. (AB)(DF) (DE)(AC) 5. Substitution postulate.AB6. AC 5 DEDF6. If the products of two pairsof factors are equal, one pairof factors can be the meansand the other the extremesof a proportion.