+468 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.
+128408 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
