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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.

 

 

 Mar 9, 2022
 #1
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0

The area of the shaded region is 11/2.

 Mar 9, 2022
 #2
avatar+2455 
+2

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: 

 

\(9 - ({30 \over13} \times {3 \over 2}) = \color{brown}\boxed{72 \over 13}\)

 Mar 9, 2022
 #5
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Can you explain, -10/13 as the slope? Oh wait, nevermind

edited by PikachuLovesKetchup  Mar 9, 2022
 #3
avatar+124598 
+2

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 !!! ]

 

cool cool cool

 Mar 9, 2022
 #6
avatar+40 
0

Cool, thanks!

 #4
avatar+40 
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Thanks guys! I appreciate it!

 Mar 9, 2022

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