AMC 10 · 2013 · #16
Grade 8 geometry-2dPick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Everything lives on the coordinate plane, so Tool #1 (Draw a Diagram) is the anchor: plot the three points, reflect them across x=8, and the shape of the overlap becomes visible. Two triangles that overlap are hard to measure directly, so Tool #16 (Change Focus) reframes the union as "both triangles added together, minus the part counted twice": union = area(T₁) + area(T₂) - overlap. Each triangle's area is a small subproblem (Tool #7), and locating the one interior corner of the overlap is a Tool #4 job — name the two slanted top edges as lines and find where they cross. The reason this works is that a reflection copies area exactly, so the only real unknown left is the shared overlap.
Reflect the three vertices
Across x=8 a point (x,y) goes to (16-x, y): A(6,5) becomes A'(10,5), C(9,1) becomes C'(7,1), and B stays put.
Mirroring over a vertical line keeps how high a point is and only swaps how far left or right of the line it sits.
8.G.A.3Draw A DiagramArea of one triangle
Shoelace on A(6,5), B(8,-3), C(9,1) gives area 8, and the mirror copy is congruent, so it has area 8 too.
The mirror image is congruent to the original, so both triangles cover exactly the same amount of space.
6.G.A.1Identify SubproblemsReframe as total minus overlap
Adding both areas counts the shared middle twice, so the union is 16 minus the overlap — only the overlap is left to find.
When two shapes are glued together, the part they share gets double-counted, so you take it back out once.
When two shapes are glued together, the part they share gets double-counted, so it is taken back out once.
▸ Why?
Adding two areas counts the overlap twice, so subtracting it once restores the honest total.
▸ Why?
The mirror image is the same size as the original, so both triangles cover exactly the same amount.
See the shared shape
C' lies on segment BA and C on segment BA', so both triangles open from B the same way — the overlap is a kite with top corner F.
The reflected triangle sits in the very same corner at B, so the shared piece is just the lower part both tops agree on.
6.G.A.3Draw A DiagramFind the top corner F
Write the top edges AC and C'A' as lines and set them equal; by symmetry they cross on x=8, giving F=(8, 7/3).
Where two lines meet is the single point that fits both equations at once.
8.EE.C.8Introduce A VariableOverlap area, then the union
Split the kite along BF, of length 16/3, into two triangles of width 1: the overlap is 16/3, so the union is 32/3.
A symmetric kite splits into two equal thin triangles, each easy to measure from the mirror line.
7.G.B.6Identify SubproblemsTo measure two overlapping copies, add both areas and subtract the shared middle once — a reflection keeps area unchanged, so all the work is just pinning down that shared piece.
- Reflect the three vertices
- Area of one triangle
- Reframe as total minus overlap
- See the shared shape
- Find the top corner F
- Overlap area, then the union