AMC 10 · 2004 · #6
Grade 8 arithmeticPick an answer.
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.
Draw the two roads from the airport
The two compass rays sit 90 degrees apart at the airport.
Northeast and northwest lean away from north by the same amount on opposite sides, so together they open up a right angle.
4.G.A.1Draw A DiagramApply the Pythagorean theorem
So the downtown distance is the hypotenuse, giving √(164).
In a right triangle the hypotenuse squared equals the sum of the two leg squares, so squaring the legs and adding gives the missing side.
Northeast and northwest lean away from north by the same amount on opposite sides, so together they open a right angle.
▸ Why?
Two angles side by side add to the angle they span together, so equal halves on either side of north total ninety degrees.
▸ Why?
In a right triangle the square on the long side equals the two squares on the legs added together.
Estimate the square root and pick the choice
Since 164 sits just under 169, the nearest whole number is 13, choice (A).
Squaring the whole numbers around the root tells you which one it sits closest to.
8.NS.A.2Eliminate PossibilitiesWhen 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