+40 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.
+2666 The slope of the line is -10/13. Multiply this by 10, and this means the line goes down -100/13 out of -130/13. This also means the line goes down 30/13 in the smaller square, meaning the area is:
\(9 - ({30 \over13} \times {3 \over 2}) = \color{brown}\boxed{72 \over 13}\)
+128474 Another method
Note that triangles ABD and FGD are similar
Call DG = x BD = 10 - x AB =10 and FG = 3
So.....by similar triangles
DG / BD = FG /AB
x / ( 10 - x) = 3 / 10 cross-multiply
10x = 3 ( 10 - x)
10x = 30 - 3x
13x = 30
x = 30 /13
The area of triangle FGD = (1/2) (FG) ( DG) = (1/2) (3) (30/13) = 90 / 26 = 45/13
So the shaded area = [ CEFG ] - [ FGD ] = 9 - 45 /13 = 72 / 13 [ as BuilderBoi found !!! ]
