Math Prize 2019 Problem 16
Source:
October 15, 2019
Math Prize for Girls
Problem Statement
The figure shows a regular heptagon with sides of length 1.
[asy]
import geometry;
unitsize(5);
real R = 1/(2 sin(pi/7));
pair A = (0, R);
pair B = rotate(360/7) * A;
pair C = rotate(360/7) * B;
pair D = rotate(360/7) * C;
pair E = rotate(360/7) * D;
pair F = rotate(360/7) * E;
pair G = rotate(360/7) * F;
pair X = B + G - A;
pair Y = (D + E) / 2;
draw(A -- B -- C -- D -- E -- F -- G -- cycle);
draw("", B -- X);
draw("", X -- G);
draw("", X -- Y);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(X);
dot(Y);
perpendicular(Y, NW, Y - A);
[/asy]
Determine the indicated length . Express your answer in simplified radical form.