Two Intersecting Chords
Source:
April 19, 2006
trigonometrygeometrygeometric transformationhomothetyratiorectangleanalytic geometry
Problem Statement
The adjoining figure shows two intersecting chords in a circle, with on minor arc . Suppose that the radius of the circle is 5, that , and that is bisected by . Suppose further that is the only chord starting at which is bisected by . It follows that the sine of the minor arc is a rational number. If this fraction is expressed as a fraction in lowest terms, what is the product ?
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=dir(200), D=dir(95), M=midpoint(A--D), C=dir(30), BB=C+2*dir(C--M), B=intersectionpoint(M--BB, Circle(origin, 1));
draw(Circle(origin, 1)^^A--D^^B--C);
real r=0.05;
pair M1=midpoint(M--D), M2=midpoint(M--A);
draw((M1+0.1*dir(90)*dir(A--D))--(M1+0.1*dir(-90)*dir(A--D)));
draw((M2+0.1*dir(90)*dir(A--D))--(M2+0.1*dir(-90)*dir(A--D)));
pair point=origin;
label("", A, dir(point--A));
label("", B, dir(point--B));
label("", C, dir(point--C));
label("", D, dir(point--D));[/asy]