AMC 10 · 2009 · #13
Grade 8 geometry-2dPick an answer.
Tool #1 (Draw a Diagram) is primary because the altitude is the only object in the problem that is not yet drawn, and drawing it turns one slanted triangle into two right triangles. Tool #7 (Identify Subproblems) then handles those two right triangles separately: each one has a hypotenuse and the shared leg 12, so each gives up its own horizontal piece by the Pythagorean theorem. Those two numbers come out completely forced — which means the entire problem is not arithmetic at all, but the placement question: where on line BC can the foot of the altitude actually sit? Tool #2 (Make a Systematic List) answers that by enumerating every position instead of assuming one, which is what turns "two possible values" into something proved rather than taken on faith. Tool #6 (Guess and Check) then builds each candidate triangle and measures its altitude, because listing a possibility is not the same as showing it exists. Tool #3 (Eliminate Possibilities) closes by reading the answer list, which quietly confirms that no single triangle can be the whole story.
Drop the altitude and name its foot
The altitude splits the picture into two right triangles.
The perpendicular from A is the one extra line that turns a slanted triangle into two right triangles you can actually measure.
4.G.A.1Draw A DiagramPythagoras fixes both pieces
Each fixes its own piece of the base.
Both right triangles hang off the same 12-long leg, so each hypotenuse pins down its own horizontal piece on its own.
Both right triangles hang off the same altitude, so each hypotenuse fixes its own horizontal piece on its own.
▸ Why?
In a right triangle the square on the long side equals the two squares on the legs added together.
▸ Why?
The altitude is the shared height of both pieces, so once the base pieces are known the area follows at once.
List every way the points can sit
The foot can land on either side, giving two lengths.
Once you know how far each vertex is from the foot, the only freedom left is which side of the foot each one is on.
7.NS.A.1Make A Systematic ListBuild both triangles and check them
Both triangles really exist, so both count.
When the foot slides outside the segment, the small right triangle gets subtracted instead of added, and the same altitude of 12 still fits.
6.G.A.1Guess And CheckAdd the two lengths
Adding gives 18, choice (D).
The choices themselves say a single triangle cannot be the answer, since no individual side length on the list would ever work.
4.NBT.B.4Eliminate PossibilitiesThe foot of an altitude does not have to land between the two corners of the side — check whether it can fall outside, because that outside picture is a whole second answer.
- Drop the altitude and name its foot
- Pythagoras fixes both pieces
- List every way the points can sit
- Build both triangles and check them
- Add the two lengths