Associating AB to A'B'
Source:
June 13, 2009
ratio
Problem Statement
In this diagram a scheme is indicated for associating all the points of segment with those of segment , and reciprocally. To described this association scheme analytically, let be the distance from a point on to and let be the distance from the associated point of to . Then for any pair of associated points, if x \equal{} a,\, x \plus{} y equals:
[asy]defaultpen(linewidth(.8pt));
unitsize(.8cm);pair D= (0,9);
pair E = origin;
pair A = (3,9);
pair P = (3.6,9);
pair B = (4,9);
pair F = (1,0);
pair G = (2.6,0);
pair H = (5,0);dot((0,0));dot((1,0));dot((2,0));dot((3,0));dot((4,0));dot((5,0));
dot((0,9));dot((1,9));dot((2,9));dot((3,9));dot((4,9));dot((5,9));
draw((D+(0,0.5))--(0,-0.5));
draw(A--H);
draw(P--G);
draw(B--F);
draw(F--H);
draw(A--B);label("",D,NW);
label("",E,NW);
label("0",(0,0),SE);
label("1",(1,0),SE);
label("2",(2,0),SE);
label("3",(3,0),SE);
label("4",(4,0),SE);
label("5",(5,0),SE);
label("0",(0,9),SE);
label("1",(1,9),SE);
label("2",(2,9),SE);
label("3",(3,9),SW);
label("4",(4,9),SE);
label("5",(5,9),SE);
label("",F,NW);
label("",G,S);
label("",H,NE);
label("",A,NW);
label("",P,N);
label("",B,NE);[/asy]
(A)\ 13a\qquad (B)\ 17a \minus{} 51\qquad (C)\ 17 \minus{} 3a\qquad (D)\ \frac {17 \minus{} 3a}{4}\qquad (E)\ 12a \minus{} 34