AMC 10 · 2009 · #10
Grade 7 geometry-2d
Pick an answer.
The diagonal BD belongs to two different triangles at once, so Tool #7 (Identify Subproblems) splits the four-sided figure into △ BCD (sides 17, 5, BD) and △ ABD (sides 5, 9, BD). Inside each triangle the third side cannot be so long or so short that the other two fail to close up — that boundary is exactly the triangle inequality, so Tool #14 (Extreme Principle) pushes each triangle to its flat, zero-area edge to read off how big and how small BD is allowed to be. Those two boundaries trap BD inside a narrow band, and Tool #3 (Eliminate Possibilities) uses the whole-number clue to pick the single integer that survives.
Cut the quadrilateral into two triangles
The diagonal cuts the shape into two triangles.
One diagonal turns a hard four-sided figure into two triangles that share that diagonal as a common side.
7.G.A.2Identify SubproblemsLower bound from the big triangle
The larger triangle gives a lower bound.
Two sides can only wrap around a long third side if together they are longer than it — otherwise they collapse into a straight line.
Two sides can only wrap around a long third side when together they reach further than it does.
▸ Why?
If they were shorter or exactly as long, the triangle would collapse into a straight line instead of closing.
▸ Why?
The two triangles put a floor and a ceiling on the diagonal, trapping it between two known walls.
Upper bound from the small triangle
The smaller gives an upper bound.
A side can be at most as long as the other two laid end to end — reach that length and the triangle flattens to a segment.
6.EE.B.8Extreme PrincipleTrap the integer
The whole-number condition traps it at 13, choice (C).
Squeeze a value between two consecutive-ish bounds and the whole-number clue leaves exactly one survivor.
6.NS.C.7Eliminate PossibilitiesA diagonal cuts a quadrilateral into two triangles; make each triangle just barely close to trap the diagonal between two numbers, then the whole-number clue picks the winner.
- Cut the quadrilateral into two triangles
- Lower bound from the big triangle
- Upper bound from the small triangle
- Trap the integer