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The square with vertices (-a, -a), (a, -a), (-a, a), (a, a) is cut by the line y = x/3 into congruent quadrilaterals. The perimeter of one of these congruent quadrilaterals divided by a equals what? Express your answer in simplified radical form.

 Nov 29, 2020
 #1
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Call the length of  the  top side of one of the quadrilaterals, 2a

 

The length of the right side of this quadilateral   (a - a/3) =  (2/3)a  =  (2a)/3

 

The length of the  bottom of the quadrilateral   =  sqrt [ ( a - -  a)^2  + (a/3 - - a/3)^2 ] =

sqrt [ 4a^2  + (4/9)a^2  ]  =   a sqrt  [ 36 + 4]/3  =  (a/3)sqrt (40)

 

The left side of the quadrilateral =  (a - - a/3) =  (4a)/3 

 

Perimeter of quadrilateral =   

 

a [ 2 + 2/3 + sqrt (40)/3  + 4/3 ]  =   a   [ 4 + sqrt (40)/3 ]  = a [ 12  + 2sqrt (10) ] / 3

 

Dividing this  by  a  =      [ 12 + 2sqrt (10)] / 3    =    4 + (2/3)sqrt (10)

 

 

cool cool cool

 Nov 29, 2020

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