AMC 10 · 2002 · #8
Grade 6 geometry-2d
Pick an answer.
Comparing B, W, and P turns into three separate area jobs, so Tool #7 (Identify Subproblems) is the backbone: find the red area, then the white area, then the blue area, and compare. Tool #1 (Draw a Diagram) supplies the trick that makes every piece countable — overlay a unit grid on the flag so the flag is 64 little squares and each tilted white square is exactly half of a 2 × 2 box. Tool #16 (Count the Complement) then gets the blue for free: instead of adding up all the blue triangles, subtract red and white from the whole flag.
Grid the whole flag
Lay a unit grid over the flag: 8 × 8 makes 64 unit squares for the three colours to share.
Covering a square with unit tiles turns 'area' into 'how many little squares,' which you can just count.
3.MD.C.7Draw A DiagramRed area P
The red square runs 4 units a side, so P = 16.
A square's area is just one side times itself, so a 4-wide square holds 16 unit tiles.
3.MD.C.7Identify SubproblemsWhite area W
Each tilted square fills half its 2 × 2 box (area 2), and there are 12 of them, so W = 24.
A diamond drawn through the midpoints of a box is always half the box, so each white square is worth 2 tiles.
A diamond drawn through the midpoints of a square is always exactly half of that square.
▸ Why?
The four corner triangles cut off are each half of their own little rectangle, so together they account for half the square.
▸ Why?
The diamond and the four corners fill the square with no gap and no overlap, so what is not corner is diamond.
Blue area B, then compare
Subtract from the whole: B = 64 - 16 - 24 = 24, so B = W, choice (A).
Since the colors tile the flag exactly, blue is just the whole minus the parts you already measured.
3.MD.C.7Change Focus Count The ComplementLay a grid over a colored picture, count each color's squares, and get the last color for free by subtracting from the whole.
- Grid the whole flag
- Red area P
- White area W
- Blue area B, then compare
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