AMC12 2017 A

AMC12 2017 A · Q4

AMC12 2017 A · Q4. It mainly tests Pythagorean theorem, Coordinate geometry.

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?

Jerry 和 Silvia 想从一个方形田地的西南角走到东北角。Jerry 先正东走然后正北走到达目标，但 Silvia 径直向东北方向直线走到目标。以下哪个选项最接近 Silvia 的行程比 Jerry 的行程短多少？

(A) 30% 30%

(B) 40% 40%

(D) 60% 60%

(E) 70% 70%

Answer

Correct choice: (A)

正确答案：（A）

Solution

If the square had side length $x$, then Jerry’s path had length $2x$, and Silvia’s path along the diagonal, by the Pythagorean Theorem, had length $\sqrt{2}x$. Therefore Silvia’s trip was shorter by $2x - \sqrt{2}x$, and the required percentage is $\frac{2x - \sqrt{2}x}{2x} = 1 - \frac{\sqrt{2}}{2} \approx 1 - 0.707 = 0.293 = 29.3\%$. The closest of the answer choices is $30\%$.

如果正方形边长为 $x$，则 Jerry 的路径长度为 $2x$，Silvia 沿对角线路径，根据勾股定理，长度为 $\sqrt{2}x$。因此 Silvia 的行程短了 $2x - \sqrt{2}x$，所需百分比为 $\frac{2x - \sqrt{2}x}{2x} = 1 - \frac{\sqrt{2}}{2} \approx 1 - 0.707 = 0.293 = 29.3\%$。答案选择中最接近的是 $30\%$ 。

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