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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. 

ZB2=FZ2+AB2(9x)2=x2+82x218x+81=x2+648164=18x17=18xx=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 89136/9=72136/9=512/9

 

Thanks! :)

 Jun 4, 2024

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