Question

Instead of walking along two adjacent sides of a rectangular field, a boy took a short cut along the diagonal and saved the distance equal to half of the longer side. Then the ratio of the shorter side to the longer side is

Answer 1

Correct Answer: 2

Concept: Use
Pythagoras theorem and distance saving.
Step 1: Let sides.
\[ \text{Shorter} = x,\quad \text{Longer} = y \]
Step 2: Distances.
\[ \text{Path along sides} = x + y \] \[ \text{Diagonal} = \sqrt{x^2 + y^2} \] Saving: \[ (x+y) - \sqrt{x^2+y^2} = \frac{y}{2} \]
Step 3: Solve.
\[ \sqrt{x^2+y^2} = x + \frac{y}{2} \] Square both sides: \[ x^2 + y^2 = x^2 + xy + \frac{y^2}{4} \] \[ y^2 - \frac{y^2}{4} = xy \] \[ \frac{3y^2}{4} = xy \Rightarrow x = \frac{3y}{4} \]
Step 4: Ratio.
\[ \frac{x}{y} = \frac{3}{4} \]
Step 5: Option analysis.

• (A) Incorrect $\times$
• (B) Correct \checkmark
• (C) Incorrect $\times$
• (D) Incorrect $\times$
• (E) Incorrect $\times$

Conclusion:
Thus, the correct answer is
Option (B).