+1234 Let $ABCD$ and $BEDF$ be two $8 \times 9$ rectangles that overlap, as shown. Find the area of the overlap.
+129771 Let the intersection of AD and FB = Z
AB = 8
Let FZ = x
Triangle ZAB is congruent to triangle ZFD
AZ = FZ
FB = 9
So ZB = FB - FZ = 9- x
By the Pythagorean Theorem
ZB^2 = FZ^2 + AB^2
(9 -x)^2 = x^2 + 8^2
x^2 - 18x + 81 = x^2 + 64
81 - 64 = 18x
17 = 18x
x = 17/18
Non-shaded area = 4 * [ ABZ ] = 4 (1/2) (AB)(AZ) = 4 (1/2) (8)(17/18) = 272/18 = 136/9
Shaded area = 8*9 - 136 / 9 = 72 - 136/9 = 512 / 9
