2021 Fall AMC 10B Problems/Problem 25
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 
[asy]
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((6,13/3)--(10,22/3));
draw((10,22/3)--(8,10));
draw((8,10)--(4,7));
draw((4,7)--(6,13/3));
label("
",(9/2,3/2),N);
label("",(11/2,9/2),S);
label("
",(1/2,9/2),E);
label("",(19/2,3/2),W);
label("",(1/2,15/2),E);
label("",(3/2,19/2),S);
label("",(5/2,13/2),N);
label("",(7/2,17/2),W);
label("",(7,43/6),W);
[/asy]
