AMC10 2010 A

AMC10 2010 A · Q7

AMC10 2010 A · Q7. It mainly tests Pythagorean theorem, Coordinate geometry.

Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles, is this last portion of her run?

Crystal有一个标记好的日常跑步路线。她从正北方向跑1英里开始。然后向东北跑1英里，再向东南跑1英里。跑步的最后一段是直线回到起点。这最后一段跑步有多远，单位英里？

(A) 1 1

(B) $\sqrt{2}$ $\sqrt{2}$

(D) 2 2

(E) $2\sqrt{2}$ $2\sqrt{2}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): When Crystal travels one mile northeast she travels $\frac{\sqrt{2}}{2}$ miles north and $\frac{\sqrt{2}}{2}$ miles east. Similarly, when she travels southeast for one mile she travels $\frac{\sqrt{2}}{2}$ miles south and $\frac{\sqrt{2}}{2}$ miles east. Just before the last portion of her run she has traveled a net of $1+\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}=1$ miles north, and $\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}=\sqrt{2}$ miles east. By the Pythagorean Theorem, the last portion of her run is $\sqrt{1^2+(\sqrt{2})^2}=\sqrt{1+2}=\sqrt{3}$ miles.

答案（C）：当 Crystal 向东北走 1 英里时，她向北走 $\frac{\sqrt{2}}{2}$ 英里、向东走 $\frac{\sqrt{2}}{2}$ 英里。类似地，当她向东南走 1 英里时，她向南走 $\frac{\sqrt{2}}{2}$ 英里、向东走 $\frac{\sqrt{2}}{2}$ 英里。在跑步的最后一段开始之前，她的净位移为：向北 $1+\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}=1$ 英里，向东 $\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}=\sqrt{2}$ 英里。由勾股定理可得，她跑步最后一段的长度为 $\sqrt{1^2+(\sqrt{2})^2}=\sqrt{1+2}=\sqrt{3}$ 英里。

Topics