use folding to construct a square with exactly 3/4 the area of the original square. Write an explanation of your methods. explain how you know the area is exactly 3/4 of the original area
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Expert's answer
2012-10-26T11:22:53-0400
Let the initial square be square 1, and the resulting square be square 2. S2 = ¾ * S1 => for sides of the square the following holds: a2 = a1*sqrt(3)/2. Fold the square in half, you will get the segment of the length a1/2. Straightenit out. Now choose the vertex of the square, and draw a circle of radius a1 / 2with the center in it. Choose the vertex neighbor to the first vertex and draw a tangent from thepoint of this vertex to the circle from the previous step. The distance between the length of this tangent will be a1* sqrt(3)/2,because the square triangle was formed with cathetus of a length a1 / 2 and hypotenuse of length a1. So, we get the segment of length a1* sqrt(3)/2. Now choose some vertex, measure this lengthalong two sizes of the square, and fold the square in the points of ends of these segments along sizes of the square.
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