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

 

 Dec 20, 2023
 #1
avatar+129435 
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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

 

cool cool cool

 Dec 20, 2023

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