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2004 AMC 10B Problem 8

2004 AMC 10B Problem 8AMC 10 Step by Step

Name: Date: September 13, 2026

Triangles: Area & Pythagorean · difficulty 1 of 5 · about 1 min · position P1-10

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?

A)
$13$
B)
$14$
C)
$15$
D)
$16$
E)
$17$

Answer

Key insight

Southwest and southeast are perpendicular directions, so the two cities and the airport form a right triangle with legs 8 and 10 and hypotenuse sqrt(164) ≈ 12.8.

Solution

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

Why this works

Compass bearings encode angles: any two of north, east, south, west (or the diagonal directions) that differ by one quarter turn are perpendicular. Once the right angle is spotted, the Pythagorean theorem finishes, and bracketing $\sqrt{164}$ between neighboring perfect squares is enough precision for a "closest to" question.

The trap

Adding 8 + 10 = 18 or otherwise treating the airport as lying on the straight path between the cities, instead of seeing the right angle at the airport.

Common mistakes

Adding 8 + 10 = 18 or otherwise treating the airport as lying on the straight path between the cities, instead of seeing the right angle at the airport.
Rounding $\sqrt{164}$ carelessly to $14$ or $15$ rather than comparing with $169 = 13^2$ .

Techniques

Bound the quantity above/below or estimate to pin it down · Set up the equation/formula and compute; no special trick needed