1986 AMC 12 第 19 题

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}

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

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

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.

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