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]