The solution is straightforward once one constructs an additional square around D with one side on the horizontal line. Then the 4 right triangles surrounding square D are identical. We then let the sides (lengths) of those triangles be x and y.
The area of square D is then equal to the area of the square around square D minus the area of the 4 right triangles surrounding square D. Working:
A = (x + y)2 - 4 (½ xy ) = x2 + 2xy + y2 - 2xy = x2 + y2
The length of a side of square D is then: Ö x2 + y 2
We next construct a square around square B so the four right triangles surrounding square B are identical - and the lengths of the sides can then be derived. This would be from the already derived lengths of the triangles and rectangles written in terms of x and y. Thus, the length of the side of square surrounding square B is:
L (b ) = 2 (x + y). The area of square B is equal to the area of the square around square B minus the area of the four right triangles surrounding square B. Working:
A b = 2(x + y)2 - 4 [½ (2x) (2y)] =
4x2 + 8xy + 4y2 - 8xy = 4x2 + 4y2
The length of the side of square B itself is then:
L b = Ö 4(x2 + 4y2) = 2Ö x2 + y 2
which is twice the length of a side of square D.
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