Question

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 :

• A. 3 : 4
• B. 2 : 3
• C. 1 : 2
• D. 1 : 4

Hint:

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 :

Answer 1

The Correct Option is A

Let longer side = L, shorter side = S. Distance along sides = L + S. Distance along diagonal = √(L^2 + S^2) .Distance saved = (L + S) - √(L^2 + S^2) = L/2. Rearrange: √(L^2 + S^2) = L + S - L/2 = L/2 + S. Square both sides: L^2 + S^2= (L/2 + S)^2 = (L^2)/4 + LS + S^2. Simplify: L^2 = (L^2)/4 + LS (Subtract S^2 from both sides). Multiply by 4: 4L^2 = L^2 + 4LSSubtract L^2: 3L^2 = 4LS. DividebyL(since L>0): 3L = 4S → S/L = 3/4 .The ratio is 3 : 4 . This matches option (1).