AMC 10 · 2010 · #7
Grade 8 geometry-2dPick an answer.
AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem is all positions and paths, so the natural tool is #1 (Draw a Diagram): drop everything onto a coordinate grid with east as +x and north as +y. The slanted miles are hard to add directly, so use Tool #7 (Identify Subproblems) to break each 45° mile into an east piece and a north piece using a 45-45-90 triangle. Add the pieces to find the endpoint, then read off a right triangle and use the Pythagorean theorem for the straight distance home.
Set up coordinates
Put the start at the origin, with east as +x and north as +y. Then every leg is just a change in x and y that you can add.
A grid turns compass directions into simple x and y moves you can add up.
6.NS.C.8Draw A DiagramThe north leg
The first mile goes due north, so only the y-coordinate changes and she stands at (0,1).
Moving straight north changes only the north coordinate.
6.NS.C.8Draw A DiagramSplit the diagonal legs
A 1-mile step at 45° is the hypotenuse of a 45-45-90 triangle, so its east and vertical pieces are each √(2)/2.
A tilted step is really two straight steps of equal size — one east and one up or down — because the 45° triangle is isosceles.
A tilted step is really two straight steps of equal size, one across and one up or down.
▸ Why?
A half right angle makes both legs equal, so the two straight parts must match.
▸ Why?
That triangle has a fixed shape, so each leg is always the same fraction of the tilted step.
Add the pieces
The two diagonals' vertical parts cancel, the east parts add to √(2), and y stays 1 — so she ends at (√(2), 1).
The two diagonals lean opposite ways vertically, so their up and down parts cancel and only the eastward push survives.
6.NS.C.8Identify SubproblemsDistance back to start
The way home is the hypotenuse of a right triangle with legs √(2) and 1, so it measures √(3) miles — choice (C).
The straight path home is the hypotenuse of a right triangle whose two legs you can read straight off the grid.
8.G.B.8Draw A DiagramPut the run on a grid and split each slanted mile into an east part and a north part — the two diagonal north parts cancel, leaving a plain right triangle with legs √(2) and 1, so the walk home is √(3) miles.
- Set up coordinates
- The north leg
- Split the diagonal legs
- Add the pieces
- Distance back to start