AMSCO'S Geometry. New York - Rye High School

Proportional Relations Among Segments Related to Triangles 503Theorem 12.12If two triangles are similar, the lengths of corresponding medians have thesame ratio as the lengths of any two corresponding sides.GivenABC ABC with the ratio ofsimilitude k :1,M is the midpointof AC, M is the midpoint of ArCr,BC a, BC a, BM m, andBM m.AMCmaBAMCmaBProvemmr 5 a ar 5 k 1StrategyHere we can use SAS to prove BCM BCM.Theorem 12.13If two triangles are similar, the lengths of corresponding angle bisectors havethe same ratio as the lengths of any two corresponding sides.GivenABC ABC with the ratio ofsimilitude k :1,E is the point atwhich the bisector of B intersectsAC, E is the point at which thebisector of B intersects ArCr,BC a, BC a, BE e, andBE = e.AECeaBAECeaBProveeer 5 a ar 5 k 1StrategyHere we can use that halves of congruent angles are congruent and AA toprove BCE BCE.The proofs of Theorems 12.12 and 12.13 are left to the student. (See exercises10 and 11.)