AMC 10 · 2009 · #23

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
area-trianglessimilar-trianglesratio-proportion identify-subproblems ↑ Prerequisites: area-triangles
📏 Medium solution 💡 3 insights
Problem
A convex quadrilateral ABCD has sides AB = 9 and CD = 12. Its diagonals AC and BD cross at a point E, and the whole diagonal AC has length 14. The triangles AED and BEC formed by the crossing diagonals have the same area. Find the length AE, the part of diagonal AC from A to the crossing point.

Pick an answer.

(A)
$\frac {9}{2}$
(B)
$\frac {50}{11}$
(C)
$\frac {21}{4}$
(D)
$\frac {17}{3}$
(E)
6

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

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.

1STEP 1

Turn equal areas into parallel sides

Adding the shared triangle DEC to both equal triangles makes ADC and BDC equal on base DC, so equal heights force AB parallel to DC.

[AED]=[BEC]→ [AED]+[DEC]=[BEC]+[DEC]→ [ADC]=[BDC]→ AB ∥ DC
2STEP 2

Read off the similar triangles

With AB parallel to DC, each diagonal cuts equal alternate angles, so triangle ABE is similar to triangle CDE.

AB ∥ DC→ ∠ BAE=∠ DCE, ∠ ABE=∠ CDE→ △ ABE ∼ △ CDE
3STEP 3

Set the similarity ratio

AB matches CD in the similarity, so the scale factor is 9 to 12, that is 3 to 4, and AE to CE has the same ratio.

AE/CE=AB/CD=9/12=3/4
4STEP 4

Split AC in the ratio 3 to 4

Write the two pieces of AC as 3t and 4t; then 7t = 14, so t = 2 and AE = 6, choice (E).

AE=3t, CE=4t, 3t+4t=14→ t=2→ AE=3t=6
Answer
6
If AE = 6 then CE = 8, and 6 + 8 = 14 matches AC, while 6 to 8 reduces to 3 to 4 as required. It also makes sense that AE is the shorter piece: AB (9) is shorter than CD (12), so the crossing point sits nearer the shorter side AB, giving AE ＜ CE, and indeed 6 ＜ 8. The value lands exactly on a listed choice with no rounding, which is what a clean AMC answer should do.
💡Key takeaway

Equal 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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