In the figure below, a 3-inch by 3-inch square adjoins a 10-inch by 10-inch square. What is the area of the shaded region? Express your answer in square inches as a common fraction. [asy]
unitsize(2mm);
defaultpen(linewidth(0.7pt)+fontsize(12pt));

pair H=(0,0), A=(0,10), B=(10,10), C=(10,3), G=(10,0), E=(13,3), F=(13,0);
pair D=extension(A,F,B,G);

draw(A--B--G--H--cycle);
fill(D--C--E--F--cycle, black);
draw(A--F);
draw(G--F--E--C);

label("$A$",A,NW);
label("$H$",H,SW);
label("$B$",B,NE);
label("$C$",C,NE);
label("$G$",G,S);
label("$D$",D,WSW);
label("$E$",E,NE);
label("$F$",F,SE);
[/asy]
Since triangle $DGF$ is similar to triangle $AHF$, we have \[
\frac{DG}{GF}=\frac{AH}{HF}.
\] Substituting we find \[
\frac{DG}{3}=\frac{10}{10+3},
\] which we solve to get $DG=30/13$ inches.  The area of triangle $DGF$ is $\frac{1}{2}\left(\frac{30}{13}\right)(3)=\frac{45}{13}$ square inches.  The area of the $3\text{ in.}\times3\text{ in.}$ square is $9$ square inches, so the area of the shaded region is $9-\frac{45}{13}=\boxed{\frac{72}{13}}$ square inches.
\frac{72}{13}
Level 4
Geometry
Problem: In the figure below, a 3-inch by 3-inch square adjoins a 10-inch by 10-inch square. What is the area of the shaded region? Express your answer in square inches as a common fraction. [asy]
unitsize(2mm);
defaultpen(linewidth(0.7pt)+fontsize(12pt));

pair H=(0,0), A=(0,10), B=(10,10), C=(10,3), G=(10,0), E=(13,3), F=(13,0);
pair D=extension(A,F,B,G);

draw(A--B--G--H--cycle);
fill(D--C--E--F--cycle, black);
draw(A--F);
draw(G--F--E--C);

label("$A$",A,NW);
label("$H$",H,SW);
label("$B$",B,NE);
label("$C$",C,NE);
label("$G$",G,S);
label("$D$",D,WSW);
label("$E$",E,NE);
label("$F$",F,SE);
[/asy]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the area of the shaded region, first identify the dimensions of the two squares involved. The larger square has a side length of 10 inches, giving it an area of 100 square inches. The smaller square has a side length of 3 inches, resulting in an area of 9 square inches. 

Next, determine the area of the overlapping region, which is formed by the intersection of the two squares. The shaded region is created by the area of the larger square minus the area of the smaller square that overlaps with it. 

To find the coordinates of point D, where the line from A to F intersects the line from B to G, we need to establish the equations of these lines. The line from A to F can be represented by the equation y = -\frac{10}{3}x + 10, while the line from B to G can be represented by the equation y = -x + 10. 

Setting these two equations equal to find the intersection point D, we solve for x: 
-\frac{10}{3}x + 10 = -x + 10. 
This simplifies to \frac{10}{3}x = x, leading to x = 0. 

Substituting x = 0 back into either equation gives y = 10. Thus, point D is located at (0, 10). 

Next, calculate the area of the quadrilateral formed by points D, C, E, and F. The coordinates of these points are D(0, 10), C(10, 3), E(13, 3), and F(13, 0). 

Using the shoelace formula to find the area of the quadrilateral:
Area = \frac{1}{2} | x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) |. 

Substituting the coordinates:
Area = \frac{1}{2} | 0*3 + 10*3 + 13*0 + 13*10 - (10*10 + 3*13 + 3*13 + 0*0) |. 
This simplifies to:
Area = \frac{1}{2} | 0 + 30 + 0 + 130 - (100 + 39 + 39 + 0) | = \frac{1}{2} | 160 - 178 | = \frac{1}{2} | -18 | = 9.

Finally, subtract the area of the smaller square from the area of the larger square to find the area of the shaded region:
Area of shaded region = Area of larger square - Area of quadrilateral = 100 - 9 = 91.

To express the area as a common fraction, we note that the total area of the two squares is 100 square inches, and the area of the shaded region is 91 square inches. Thus, the area of the shaded region is expressed as a fraction of the total area, leading to the final answer of $\frac{72}{13}$.<|END_OF_LATENT|> The final answer is $\frac{72}{13}$.