AMSCO'S Geometry. New York - Rye High School

Dilations 497Slope ofAB 5 d b 2 2 a05 b 2 daSlope of ArBr 5 kb kd 2 2 ka05 k dk A b 2 a B5 b 2 daTherefore, AB ArBr.Slope ofAC 5 c e 2 2 0a5 c 2 eaSlope of ArCr 5 kc ke 2 2 ka05 k ek A c 2 a B5 c 2 eaTherefore, AC ArCr.Slope ofBC 5 d 2 eb 2 ckd 2 keSlope of BrCr 5 kb 2 kc5 k k A d b 2 2 c e B5 d b 2 2 ceTherefore, BC BrCr.We have shown that AB ArBr and AC ArCr. Therefore, because they arecorresponding angles of parallel lines:mOAB mOABmOAC mOACmOAB mOAC mOAB mOACmBAC mBACIn a similar way we can prove that ACB ACB, and soABC ABC by AA. Therefore, under a dilation, angle measure is preservedbut distance is not preserved. Under a dilation of k, distance is changedby the factor k.We have proved the following theorem:Theorem 12.8Under a dilation, angle measure is preserved.We will now prove that under a dilation, midpoint and collinearity arepreserved.Theorem 12.9Under a dilation, midpoint is preserved.Proof: Under a dilation D k:MA(a, c) → A(ka, kc)B(b, d) → (kb, kd)A a 1 b2 , c 1 d2 B→ MThe coordinates of the midpoint ofare:ka 1 kb kc 1 kdA 2 , 2 BorA k a 1 b2 , k c 1 d2 BArBrA k a 1 b2 , k c 1 d2 BTherefore, the image of M is the midpointof the image of AB, and midpoint ispreserved.yOAAMBMBx