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A square sheet of paper has area \( 6 \text{cm}^2\) The front is white and the back is black. When the sheet is folded so that point A rests on the diagonal as shown, the visible black area is equal to the visible white area. How many centimeters is A from its original position? Express your answer in simplest radical form.

 

 Oct 6, 2018
 #1
avatar+96302 
+1

The sides of the square must be √6 cm  each

 

Let x be the distance between the legs of the black triangle and the side of the square

 

The white area is composed of a square of sides of  x  and two rectangles with dimensions of x and [ √6 - x ]   

 

So.....the white area  =    x^2 + 2x ( √6 - x )

 

And the area of the  black triangle  =  ( √6 - x )^2 / 2  = [ 6 - [2√6]x  + x^2] / 2

 

Since these areas are equal  we have that

 

x^2 + 2x ( √6 - x )  = [ 6  - [2√6]x + x^2] / 2

 

2 [ x^2  + 2x (√6 - x) ] = 6 - [2√6]x + x^2 

 

2x^2 + 4x(√6 - x ]   = 6 - [2√6]x  + x^2

 

x^2 + [4√6] x - 4x^2  = 6 - [2√6]x

 

-3x^2 + [ 6√6]x   = 6

 

3x^2  - (6√6]x  = -6

 

x^2 - [2√6]x = -2     complete the square

 

x^2  - [2√6]x  +  6  =  -2 + 6

 

( x - √6)^2  = 4    take both roots

 

x - √6  = 2        or     x - √6  = -2

 

x = √6 + 2       or     x  =  √6  - 2

(reject)                           (accept)

 

So  ...let the original position of A   = (0, 0)

 

And the new position  of A  =  ( √6  - ( √6  - 2) ,  √6 - ( √6 - 2) )  = ( 2 , 2)

 

And its distance from its original position is

 

√ [ 2^2  +  2^2  ]  =

 

√8  =

 

2√2

 

 

 

cool cool cool

 Oct 6, 2018
edited by CPhill  Oct 6, 2018
 #2
avatar+464 
0

Thank you CPhill

 Oct 8, 2018

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