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Let $ABCD$ and $BEDF$ be two $8 \times 9$ rectangles that overlap, as shown.  Find the area of the overlap.

 

 Jun 4, 2024
 #1
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Ok, let's first let the intersection of AD and FB be Z. 

We know AB = 8. 

Lets let FZ = x

 

From the problem, we can tell that triangles ZFD and ZAB are congruent triangles. From this, we know FB = 9. 

ZB = FB - FZ = 9 - x

 

Now, let's use the awesome pythaogrean theorem. 

\(ZB^2 = FZ^2 + AB^2 \\ (9 -x)^2 = x^2 + 8^2 \\ x^2 - 18x + 81 = x^2 + 64 \\ 81 - 64 = 18x \\ 17 = 18x \\ x = 17/18\)

 

Now, we can find the non-shaded area. 

\(4 * [ ABZ ] = 4 (1/2) (AB)(AZ) = 4 (1/2) (8)(17/18) = 272/18 = 136/9\)

 

Now, the shaded area is \(8*9 - 136 / 9 = 72 - 136/9 = 512 / 9\)

 

Thanks! :)

 Jun 4, 2024

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