AMC 10 · 2004 · #8
Grade 6 geometry-2d
Pick an answer.
The two triangles we are asked about, △ ADE and △ BDC, are the awkward slivers left after the diagonals cross — finding either one directly would need the location of D. The unlock is to stop chasing the slivers and look at the whole triangles they belong to. Point D sits on BE, so △ ADE is just △ ABE with the shared corner △ ABD removed; likewise △ BDC is △ ABC with that same △ ABD removed. That is a classic change of focus (Tool #16): add the common overlap back to both pieces instead of computing the pieces. The two full triangles are easy right triangles, so their areas come straight from the legs (Tool #1 to see which segments are the legs). Naming the shared overlap with a single variable (Tool #4) makes the cancellation visible: the unknown region drops out when we subtract, so we never need to find D at all.
Area of each whole right triangle
Both are right triangles, so their whole areas are 16 and 12.
A right angle hands you the base and height for free — the two legs are perpendicular, so no extra work is needed to find a height.
6.G.A.1Draw A DiagramName the shared overlap
They overlap in one corner region; call it x rather than measuring it.
When a quantity is hard to find but appears in both things you are comparing, name it and let it cancel later rather than fighting to compute it.
6.EE.B.6Introduce A VariableWrite each target triangle as whole minus overlap
Each target sliver is its whole triangle minus that region: 16-x and 12-x.
Adding the same overlap to each sliver rebuilds a full, easy triangle — so the sliver is just the full area minus that shared overlap.
6.G.A.1Change Focus Count The ComplementSubtract and watch the unknown cancel
Subtracting cancels the unknown, leaving 4, choice (B).
Subtracting two amounts that share the same unknown makes it vanish, leaving only the difference of the parts you actually know.
Subtracting the two areas makes the shared overlap vanish, leaving only the parts that are known.
▸ Why?
The same overlap is taken from each whole triangle, so it cancels the moment one is subtracted from the other.
▸ Why?
Each whole triangle is right-angled, so its two legs serve as base and height and its area needs no extra measuring.
When two shapes overlap in the same messy piece, don't fight to measure that piece — give it a name and compare the whole shapes instead. The shared part cancels, and here the answer is just 16 - 12 = 4.
- Area of each whole right triangle
- Name the shared overlap
- Write each target triangle as whole minus overlap
- Subtract and watch the unknown cancel