2021 Fall AMC 10B Problems/Problem 25

Problem

A rectangle with side lengths and a square with side length and a rectangle are inscribed inside a larger square as shown. The sum of all possible values for the area of can be written in the form , where and are relatively prime positive integers. What is

=Solution

We see that the polygon bounded by the small square, large square, and rectangle of known lengths is an isosceles triangle. Let’s draw a perpendicular from the vertex of this triangle to its opposing side;

size(8cm);
draw((0,0)--(10,0));
draw((0,0)--(0,10));
draw((10,0)--(10,10));
draw((0,10)--(10,10));
draw((1,6)--(0,9));
draw((0,9)--(3,10));
draw((3,10)--(4,7));
draw((4,7)--(1,6));
draw((0,3)--(1,6));
draw((1,6)--(10,3));
draw((10,3)--(9,0));
draw((9,0)--(0,3));
draw((1,6)—-(0,6));
draw((6,13/3)--(10,22/3));
draw((10,22/3)--(8,10));
draw((8,10)--(4,7));
draw((4,7)--(6,13/3));
label("$3$",(9/2,3/2),N);
label("$3$",(11/2,9/2),S);
label("$1$",(1/2,9/2),E);
label("$1$",(19/2,3/2),W);
label("$1$",(1/2,15/2),E);
label("$1$",(3/2,19/2),S);
label("$1$",(5/2,13/2),N);
label("$1$",(7/2,17/2),W);
label("$R$",(7,43/6),W);
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Solution in progress ~KingRavi

Video Solution

https://www.youtube.com/watch?v=5mPvkipCvhE

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.