A man walked diagonally across a square lot. Approximately, what was the percent saved by not walking along the edges?

### Concept & Path Optimization in Geometry Walking along the edges involves traveling two sides of the square, whereas walking diagonally involves traversing the hypotenuse of the right-angled triangle formed by the sides. $$ \text{Diagonal} = a\sqrt{2} $$ ### Step-by-Step Solution * Let the side of the square lot be $a$. * If the man walks along the edges from one corner to the opposite, he covers two sides. Distance = $a + a = 2a$. * If he walks diagonally, the distance is $a\sqrt{2}$. Since $\sqrt{2} \approx 1.414$, the diagonal distance is approximately $1.414a$. * Distance saved = $2a - 1.414a = 0.586a$. * Percentage saved = $(\frac{\text{Distance Saved}}{\text{Original Distance}}) \times 100$. * Percentage saved = $(\frac{0.586a}{2a}) \times 100 = 0.293 \times 100 = 29.3\%$. * Approximately, this is 30%. ### Exam Strategy & Shortcut Memorize that the ratio of the diagonal to the sum of two sides of a square is $\sqrt{2} : 2$, or about $1.41 : 2$. The saving is $(2 - 1.41) / 2 = 0.59 / 2 \approx 0.30$ or $30\%$. Knowing this standard result saves you from doing calculations on the spot. ### Common Pitfall A common mistake is calculating the percentage relative to the new diagonal path (dividing by $1.414a$) rather than the original longer path (dividing by $2a$). ### Final Answer Therefore, the correct answer is **30**.