Shortcut Across A Field

Geometry Level 2

Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal and saved distance equal to half of the longer side.

What is the ratio of the length of the field's shorter side to the length of its longer side?

2 3 \dfrac23 3 4 \dfrac34 1 4 \dfrac14 1 2 \dfrac12

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Amish Garg by Amish Garg , 20, India

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1 solution

Tanvir Hasan
Jan 29, 2016

Let a ≤ b a\le b be the side lengths. Using the pythagorean theorem , the diagonal has length a 2 + b 2 . \sqrt{a^2 + b^2}.

From the given information, the length of the diagonal is ( a + b ) − b 2 = a + b 2 . (a+b) - \frac{b}{2} = a + \frac{b}{2}. Thus, we need a 2 + b 2 = a + b 2 . \sqrt{a^2+b^2} = a + \frac{b}{2}.

This gives 4 a 2 + b 2 + 4 a b = 4 ( a 2 + b 2 ) , 4a^2 + b^2 + 4ab = 4(a^2+b^2),

so 3 b 2 = 4 a b , 3b^2 = 4ab,

so a b = 3 4 . \frac{a}{b} = \frac{3}{4}.

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