If you fold an A4 sheet of paper in half (the fold must be parallel to the shorter sides) you will have divided the rectangle into two similar rectangles (A5 size). Using this fact alone what is the ratio of length to the width of the paper? (No measuring allowed!)
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Let the original A4 sheet of paper be of length x and width y.
The smaller rectangle (A5) will have sides of length y and width x/2.
As the rectangles are similar the ratio of the sides must be the same.
x/y = y/(x/2)
x2 = 2y2
x = √2 y
So the ratio of the sides is 1:√2
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