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. Then the ratio of the shorter side to the longer side is:

• A. \( \frac{1}{2} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{3}{4} \)

Hint:

When dealing with diagonals in rectangles, use the Pythagorean theorem and relate the sides to the diagonal.

Answer 1

The Correct Option is B

Let the shorter side of the rectangle be \( x \) and the longer side be \( y \). The diagonal forms a right triangle, and by the Pythagorean theorem: \[ \text{Diagonal} = \sqrt{x^2 + y^2} \] According to the problem, the boy saved half of the longer side, so: \[ y - \sqrt{x^2 + y^2} = \frac{y}{2} \] Solving this equation will give the ratio of \( x \) to \( y \). The correct ratio is \( \frac{2}{3} \).