AMC 10 · 2013 · #16

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation
centroid-2-to-1integer-pythagorean-triplesarea-triangles identify-subproblemspattern-recognition ↑ Prerequisites: pythagorean-theoremarea-triangles
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
In triangle ABC, the medians AD and CE meet at point P. The three small segments near P measure PE=1.5, PD=2, and DE=2.5. Find the area of quadrilateral AEDC (the four corners A, E, D, C in order).

Pick an answer.

(A)
13
(B)
13.5
(C)
14
(D)
14.5
(E)
15

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The figure looks busy, so tool #7 (Identify Subproblems) breaks AEDC into small, familiar pieces: first find the missing half-lengths of the medians, then check the angle at P, then build the area. Tool #5 (Look for a Pattern) spots that the little triangle PED has sides 1.5, 2, 2.5 — a scaled 3-4-5 right triangle — which reveals a right angle. Tool #1 (Draw a Diagram) keeps track of which segments are the diagonals of AEDC and how they cross at P.

1STEP 1

Use the centroid 2:1 split

The medians meet at the centroid P, which cuts each in a 2:1 split with the long part at the vertex: AP=4, CP=3.

AP = 2 · PD = 2 · 2 = 4, CP = 2 · PE = 2 · 1.5 = 3
2STEP 2

Spot the right angle at P

Triangle PED has sides 1.5, 2, 2.5 — a halved 3-4-5 — so by the Pythagorean converse the medians cross at 90° at P.

1.5² + 2² = 2.25 + 4 = 6.25 = 2.5² → ∠ EPD = 90°
3STEP 3

Name the diagonals of AEDC

Going around A, E, D, C, the diagonals are the medians themselves: AD=6 and EC=4.5, and they are perpendicular.

AD = AP+PD = 4+2 = 6, EC = EP+PC = 1.5+3 = 4.5
4STEP 4

Area from perpendicular diagonals

Perpendicular diagonals mean the area is half their product: half of 6 times 4.5 gives 13.5, choice (B).

[AEDC] = 1/2 · AD · EC = 1/2 · 6 · 4.5 = 13.5 → (B)
Answer
13.5
The answer 13.5 sits right in the middle of the choices 13 to 15, so it is a sensible size. Quick cross-check: the four right triangles have legs (4,3), (4,1.5), (2,3), (2,1.5), giving areas 6, 3, 3, 1.5, which total 13.5 — matching the 1/2 d₁ d₂ shortcut exactly. The right angle at P is guaranteed by the 3-4-5 pattern, so no measuring is needed.
💡Key takeaway

The two medians are the diagonals of AEDC; they cross at a right angle, so the area is just half the product of the two full median lengths.

  • Use the centroid 2:1 split
  • Spot the right angle at P
  • Name the diagonals of AEDC
  • Area from perpendicular diagonals

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