Purple Comet 2009 MS Problem 14
Source:
April 21, 2012
geometryrectanglesimilar triangles
Problem Statement
Rectangle measures by . Eighteen points (including and ) are marked on the diagonal dividing the diagonal into congruent pieces. Twenty-two points (including A and B) are marked on the side dividing the side into congruent pieces. Seventeen non-overlapping triangles are constructed as shown. Each triangle has two vertices that are two of these adjacent marked points on the side of the rectangle, and one vertex that is one of the marked points along the diagonal of the rectangle. Only the left of the congruent pieces along the side of the rectangle are used as bases of these triangles. Find the sum of the areas of these triangles.[asy]
size(200);
defaultpen(linewidth(0.8));
pair A=origin,B=(21,0),C=(21,12),D=(0,12);
path P=origin;
draw(A--B--C--D--cycle--C);
for (int r = 1; r <= 17;++r) {
P=P--(21*r/17,12*r/17)--(r,0);
}
P=P--cycle;
filldraw(P,gray(0.7));
label("",A,SW);
label("",B,SE);
label("",C,NE);
label("",D,NW);
[/asy]