Instead of walking along two adjacent sides of a rectangular field, a boy took a short cut along the diagonal and saved a distance equal to half the longer side. Ratio of the length of shorter side to that of the longer side is :
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Instead of walking along two adjacent sides of a rectangular field, a boy took a short cut along the diagonal and saved a distance equal to half the longer side. Ratio of the length of shorter side to that of the longer side is :
Let the longer side be \(L\) and the shorter side be \(S\).
The distance along the sides is \(L + S\).
The distance along the diagonal is \(\sqrt{L^2 + S^2}\).
The distance saved is \((L + S) - \sqrt{L^2 + S^2}\).
It's stated to be \(L/2\).
Rearranging, \(\sqrt{L^2 + S^2} = L + S - L/2 = L/2 + S\).
Squaring both sides: \(L^2 + S^2 = (L/2 + S)^2 = (L^2)/4 + LS + S^2\).
Simplifying: \(L^2 = (L^2)/4 + LS\).
Multiply by 4: \(4L^2 = L^2 + 4LS\).
Subtract \(L^2\): \(3L^2 = 4LS\).
Divide by \(L\) (since \(L > 0\)): \(3L = 4S \rightarrow S/L = 3/4\).
The ratio is \(3 : 4\). This corresponds to option (1).