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Question

Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal of the field and saved a distance equal to 12 the longer side. The ratio of the shorter side of the rectangle to the longer side was:

Accepted Answer

Let x<y be the sides of the rectangle. The length of the diagonal is x^2+y^2, and the length of the two adjacent sides is x+y. Then the distance the boy saves is x+y-x^2+y^2. Setting this equal to 12y, we have x+y-x^2+y^2=12y x+12y=x^2+y^2 x^2+xy+14y^2=x^2+y^2 xy=34y^2 xy=34, so the answer is 34.