Question

一个公园呈正六边形，每边长 $2$ 千米。Alice 从一个顶点出发，沿公园周界走了 $5$ 千米。此时她与起点相距多少千米？

A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?

$\sqrt{13}$

$\sqrt{14}$

$\sqrt{15}$

$\sqrt{16}$

$\sqrt{17}$

答案：A

知识点：regular hexagon displacement Pythagorean theorem

难度评级：1620

小提示：

这段路程由两条完整的边和下一条边的一半组成

The walk consists of two complete sides and half of the next side

大提示：

将三个有向线段分解为水平分量和竖直分量

Resolve the three directed segments into horizontal and vertical components

解答：

令第一条边沿水平方向。三段有向路程对应的向量为 $\begin{gathered} (2,0),\qquad (1,\sqrt3),\\ \left(-\frac12,\frac{\sqrt3}{2}\right)\text{。} \end{gathered}$ 它们的和为 $(\frac{5}{2},\frac{3\sqrt3}{2})$ 。其长度的平方为 $\left(\frac52\right)^2+\left(\frac{3\sqrt3}{2}\right)^2=13\text{。}$ 因此距离为 $\sqrt{13}$ 千米。

所以正确答案是 A。

Choose the first side in the horizontal direction. The three directed portions of the walk have vectors $\begin{gathered} (2,0),\qquad (1,\sqrt3),\\ \left(-\frac12,\frac{\sqrt3}{2}\right). \end{gathered}$ Their sum is $(\frac{5}{2},\frac{3\sqrt3}{2}).$ Its squared length is $\left(\frac52\right)^2+\left(\frac{3\sqrt3}{2}\right)^2=13.$ The distance is therefore $\sqrt{13}$ km.

Thus the correct answer is A.

1986 AMC 12 第 19 题

答案：A

难度评级：1620

小提示：

大提示：

解答：

其他年份的第 19 题