AMC10 2017 A

AMC10 2017 A · Q7

AMC10 2017 A · Q7. It mainly tests Pythagorean theorem, Area & perimeter.

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?

杰瑞和西尔维娅想从一个方形田地的西南角走到东北角。杰瑞先正东走然后正北走到达目标，但西尔维娅径直向东北方向直线走到目标。以下哪个是最接近西尔维娅的行程相比杰瑞的行程少多少？

(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\n\[\frac{2x - \sqrt{2}\,x}{2x} = 1 - \frac{\sqrt{2}}{2} \approx 1 - 0.707 = 0.293 = 29.3\%.\]\nThe closest of the answer choices is 30%.

如果正方形边长为\(x\)，则杰瑞的路径长度为\(2x\)，西尔维娅沿对角线路径，根据勾股定理，长度为\(\sqrt{2}\,x\)。因此西尔维娅的行程短了\(2x - \sqrt{2}\,x\)，所需百分比为\n\[\frac{2x - \sqrt{2}\,x}{2x} = 1 - \frac{\sqrt{2}}{2} \approx 1 - 0.707 = 0.293 = 29.3\%\]\n最接近的答案选项是30%。

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