1986 AMC 12 Problem 19

Question

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}

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

Resolve the three directed segments into horizontal and vertical components

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.

Problem 19 in other years