AMC 10 · 2004 · #8

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
pythagorean-theoremestimation physical-representation ↑ Prerequisites: pythagorean-theorem
📏 Short solution 💡 2 insights
Problem
An airport sits 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Using these two compass distances, find which whole number of miles is closest to the straight-line distance between the two downtowns.

Pick an answer.

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

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The problem is really about directions, so the first move is to place the airport on a diagram and draw the two segments to the downtowns (Tool #1). Reading the compass words carefully (Tool #17) shows the segment to St. Paul points northeast and the segment to Minneapolis points northwest — and those two directions are perpendicular. That turns the picture into a right triangle whose legs are 8 and 10 and whose hypotenuse is the distance we want, so the Pythagorean theorem finishes it. Because the true distance is not a whole number, Tool #3 (Eliminate Possibilities) is used at the end to pick the closest choice.

1STEP 1

Draw the two roads from the airport

Put the airport at a compass center: St. Paul lies 8 miles northeast, Minneapolis 10 miles northwest, so the two rays meet at 90°.

45° + 45° = 90°
2STEP 2

Apply the Pythagorean theorem

The two downtowns are joined by the hypotenuse: d2=82+102=164d^2 = 8^2 + 10^2 = 164, so the distance is 164\sqrt{164} miles.

d = √(8² + 10²) = √(64 + 100) = √(164)
3STEP 3

Estimate the square root and pick the choice

122=14412^2 = 144 and 132=16913^2 = 169, and 12.82=163.8412.8^2 = 163.84, so 164≈12.8\sqrt{164}\approx 12.8 — the nearest choice is (A) 13.

12² = 144 ＜ 164 ＜ 169 = 13², 12.8² = 163.84 → √(164)≈ 12.8 → (A) 13
Answer
13
The hypotenuse of a right triangle must be longer than either leg but shorter than their sum, so the distance sits between 10 and 8+10=18 miles. The estimate √(164)≈ 12.8 falls in that window and is only a little more than the longer leg of 10, which is exactly what a right triangle with legs 8 and 10 should give. Rounding to the nearest listed value gives 13, choice (A).
💡Key takeaway

When two compass directions meet at a right angle, the straight-line distance across is the hypotenuse — square the two legs, add, and take the square root.

  • Draw the two roads from the airport
  • Apply the Pythagorean theorem
  • Estimate the square root and pick the choice

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