Index Cards :D
Source: 2022 AMC 10A #10
November 11, 2022
AMC 10AMC2022 AMC 10a2022 AMCgeometryarea
Problem Statement
Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be centimeters, as shown below. What is the area of the original index card?[asy]
unitsize(0.6 cm);pair A, B, C, D, E, F, G, H;
real x, y;
x = 9;
y = 5;A = (0,y);
B = (x - 1,y);
C = (x - 1,y - 1);
D = (x,y - 1);
E = (x,0);
F = (1,0);
G = (1,1);
H = (0,1);draw(A--B--C--D--E--F--G--H--cycle);
draw(interp(C,G,0.03)--interp(C,G,0.97), dashed, Arrows(6));
draw(interp(A,E,0.03)--interp(A,E,0.97), dashed, Arrows(6));label("", (B + C)/2, W);
label("", (C + D)/2, S);
label("", interp(A,E,0.3), NE);
label("", interp(G,C,0.2), SE);
[/asy]