2004 AMC 10B Problem 8

1. Draw it from the airport's point of view. The airport is southwest of St. Paul, so St. Paul is northeast of the airport. The airport is southeast of Minneapolis, so Minneapolis is northwest of the airport.
2. Northeast and northwest are a quarter turn apart, which is $90^\circ$. So the airport and the two cities form a right triangle, with the right angle at the airport.
3. The legs are $8$ and $10$ miles. The distance between the cities is the hypotenuse:
$$\sqrt{8^2 + 10^2} = \sqrt{64 + 100} = \sqrt{164}.$$
4. We only need the closest whole number. $12^2 = 144$ and $13^2 = 169$. Since $164$ is much closer to $169$, $\sqrt{164}$ is a little under $13$ (about $12.8$).

The answer is $\boxed{\textbf{(A)}\ 13}$.

Compass directions hide angles. Directions that are a quarter turn apart, like northeast and northwest, are perpendicular, so they give a right triangle. For a "closest to" question you do not need the exact square root; just find the two perfect squares it sits between.

From the airport, St. Paul lies to the northeast (the airport is southwest of it) and Minneapolis lies to the northwest. Northeast and northwest are $90^\circ$ apart, so the airport is the right-angle vertex of a right triangle whose legs are $8$ and $10$ miles.

The distance between the two downtowns is the hypotenuse:
$$\sqrt{8^2 + 10^2} = \sqrt{64 + 100} = \sqrt{164}.$$
Since $12^2 = 144$ and $13^2 = 169$, and $164$ is only $5$ short of $169$, the length is about $12.8$ miles, closest to $13$.

The answer is $\boxed{\textbf{(A)}\ 13}$.