2004 AMC 10B Problem 8

P1-10~1 minPrint

Triangles: Area & Pythagorean Bound the quantity above/below or estimate to pin it down Set up the equation/formula and compute; no special trick needed

Minneapolis-St. Paul International Airport is $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?

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Key insight

Seen from the airport, one city is northeast and the other northwest. That is a right angle, so use the Pythagorean theorem with legs 8 and 10.

Solution

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 $\boxed{\textbf{(A)}\ 13}$ .

Why this works

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.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(A)}\ 13}$ .

Review in the AoPS books

Introduction to Geometry, Chapter 4: Perimeter and Area (§4.2)
Introduction to Geometry, Chapter 6: Right Triangles (§6.1, §6.2, §6.3)
Introduction to Geometry, Chapter 10: Geometric Inequalities (§10.3)