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Square $ABCD,$ shown here, has side length $6$ units. Points $P$ and $Q$ are located on sides $AD$ and $BC,$ respectively, with $AP = BQ = 1$ unit. Triangles $ACP$ and $BDQ$ overlap in the square to form the shaded quadrilateral. What is the area of the shaded quadrilateral? Express your answer as a common fraction. 

 Feb 11, 2024
 #1
avatar+129852 
+1

 

EF  parallel to QB

So triangle DEF is similar to  triangle DQB

 F is the midpoint of DB (the diagonal of the square)

So

DF / DB   =    EF / QB

 

(1/2)DB  / DB  =  EF / QB

 

(1/2) = EF / 1

 

EF =1/2

 

And EF is the diagonal of the square formed by the overlap

 

So.....the side of the squre  =   (1/2)(1/sqrt2)  =  1/ sqrt 8

 

Area of the quadrialeral   =  (1/sqrt8)^2  =   1/8

 

 

cool cool cool

 Feb 11, 2024

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