AMC 10 · 2008 · #25
Grade 10 geometry-2dPick an answer.
Tool #1 (Draw a Diagram) is primary because the whole problem is unlocked by two auxiliary lines, not by computation. Drop the perpendiculars from A and B to CD and the two legs become right triangles; then extend each angle bisector until it hits the far parallel side, and an isosceles triangle appears that pins P and Q down exactly. Tool #4 (Introduce a Variable) sets coordinates so the two leg lengths turn into two equations in the foot position a and the height h — and, importantly, so that no assumption has to be made about where the feet land. Tool #7 (Identify Subproblems) splits the work into three independent pieces that can each be finished cleanly: find the height, locate P and Q, then assemble areas. Tool #15 (Organize Information in More Ways) supplies the independent second route in the review — instead of cutting two triangles off the trapezoid, slice the hexagon along PQ and add two smaller trapezoids. Tool #3 (Eliminate Possibilities) gives a cheap guard: the height is bounded by the shorter leg, which kills one choice before any real work happens.
Set coordinates on the long parallel side
Coordinates on the long side make everything writable.
Putting the long parallel side on the x-axis makes "height of the trapezoid" and "y-coordinate" the same thing.
10.G-GPE.B.4Introduce A VariableTurn the two legs into two equations
The two legs give two equations.
A slanted segment of known length is a right triangle in disguise: horizontal run and vertical rise squared add to its square.
8.G.B.8Draw A DiagramDifference of squares gives the height
Subtracting them gives the height directly.
The two leg equations differ only in one square, so subtracting them trades a messy pair of square roots for a single linear equation.
9.A-SSE.A.2Introduce A VariableThe bisectors at A and D meet at a right angle
Supplementary angles make the two bisectors meet at a right angle.
Parallel sides force the two corner angles on a leg to fill a straight angle, so their halves fill a right angle.
Parallel sides force the two corner angles on one leg to fill a straight angle, so their halves fill a right angle.
▸ Why?
A line crossing two parallels makes the angles on the same side add up rather than match.
▸ Why?
Those two angles together lie along one straight line, so their total is a straight angle.
Extend the bisector to find P exactly
Extending a bisector locates each meeting point exactly.
Push an angle bisector until it lands on the opposite parallel side and it always builds an isosceles triangle, which hands you the exact midpoint for free.
10.G-CO.C.10Draw A DiagramCheck the hexagon is two clean corner cuts
The hexagon is the trapezoid with two corners cut off, and they do not overlap.
The two bisector feet land far apart on the long side, so the corners being trimmed never touch each other.
10.G-GPE.B.4Identify SubproblemsAdd up the three areas
Subtracting both gives 30√(3), choice (C).
A median splits a triangle into halves of equal area, so the midpoint fact converts straight into an area fact.
6.G.A.1Identify SubproblemsPush an angle bisector until it hits the opposite parallel side: it always builds an isosceles triangle, and the point you were chasing turns out to be an exact midpoint.
- Set coordinates on the long parallel side
- Turn the two legs into two equations
- Difference of squares gives the height
- The bisectors at A and D meet at a right angle
- Extend the bisector to find P exactly
- Check the hexagon is two clean corner cuts
- Add up the three areas
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