2005 AMC 8 第 7 题

Question

比尔先向南走

\frac{1}{2}

英里，再向东走

\frac{3}{4}

英里，最后又向南走

\frac{1}{2}

英里。沿直线计算，他离起点有多少英里？

Bill walks

\frac{1}{2}

mile south, then

\frac{3}{4}

mile east, and finally

\frac{1}{2}

mile south. How many miles is he, in a direct line, from his starting point?

1

1 \frac{1}{4}

1 \frac{1}{2}

1 \frac{3}{4}

2

将两段向南的路合并成一个竖直位移

Combine the two southward parts into one vertical displacement.

用直角边

1

和

\frac{3}{4}

应用勾股定理

Use the Pythagorean theorem with legs

1

and

\frac{3}{4}

.

所求是对角线长度。使用勾股定理：

\sqrt{\left(\dfrac{1}{2} + \dfrac{1}{2}\right)^2 + \dfrac{3}{4}^2}

= \sqrt{1 + \dfrac{9}{16}} = \sqrt{\dfrac{25}{16}} = \dfrac{5}{4}

所以正确答案是 B。

Note that what we want is the length of the diagonal. We can use the Pythagorean theorem to find this.

\sqrt{\left(\dfrac{1}{2} + \dfrac{1}{2}\right)^2 + \dfrac{3}{4}^2}

= \sqrt{1 + \dfrac{9}{16}} = \sqrt{\dfrac{25}{16}} = \dfrac{5}{4}

Thus, B is the correct answer.

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