AMC 10 · 2009 · #20
Grade 8 geometry-2dPick an answer.
No picture is given, so I draw the quadrilateral with its two diagonals. The crossing point E splits the figure into four triangles, and the equal-area clue is a statement about two of them. On the diagram I can add a shared triangle to both, which turns the area clue into a parallel-sides fact. Parallel sides then hand me a pair of similar triangles, and the side ratio finishes the problem.
Turn equal areas into parallel sides
Adding a common piece turns equal areas into parallel sides.
Two triangles on the same base with equal area must have the same height, so their far vertices lie on a line parallel to that base.
Two triangles on the same base with equal area must have the same height, so their far vertices lie on a parallel line.
▸ Why?
Triangles on one base between the same parallels always have the same area, and the converse pins the parallel.
▸ Why?
An area is half the base times the height, so with the base shared the areas match exactly when the heights do.
Read off the similar triangles
Parallel sides give similar triangles at the crossing.
Parallel sides cut by the diagonals create matching angles, and matching angles mean the triangles are scaled copies.
8.G.A.5Draw A DiagramSet the similarity ratio
The known sides set the ratio.
Similar triangles shrink every matching side by the same factor, so one known pair fixes the ratio of the others.
8.G.A.4Introduce A VariableSplit AC in the ratio 3 to 4
Splitting the diagonal in that ratio gives 6, choice (A).
When a whole is split in a known ratio, count the equal parts and share the total among them.
7.RP.A.3Introduce A VariableEqual triangle areas often hide a pair of parallel sides, and parallel sides give similar triangles whose matching sides share one ratio.
- Turn equal areas into parallel sides
- Read off the similar triangles
- Set the similarity ratio
- Split AC in the ratio 3 to 4
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