AMC 10 · 2007 · #14

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation
equilateral-trianglearea-triangles double-counting ↑ Prerequisites: area-triangles
📏 Medium solution 💡 3 insights
Problem
A point inside an equilateral triangle has three known perpendicular distances to the sides. Find the side length.

Pick an answer.

(A)
4
(B)
$3\sqrt{3}$
(C)
6
(D)
$4\sqrt{3}$
(E)
9
How to solve
Strategy Identify Subproblems

The three perpendiculars beg to be joined into a picture: connect P to each vertex and the equilateral triangle splits into three smaller triangles (Tool #1, then Tool #7). Because each perpendicular is the height of one small triangle onto a side of length s=AB, the three areas are easy to write down and add. Setting that total equal to the area of the whole equilateral triangle — computed with one variable s (Tool #4) — turns the geometry into a single equation to solve (Tool #13). This area-splitting idea is exactly Viviani's Theorem in disguise.

1STEP 1

Split the triangle from P

Joining the point to the corners splits the triangle in three.

[ABC]=[PAB]+[PBC]+[PCA]
2STEP 2

Add the three small areas

Every piece shares the same base, so their areas add to three times the side.

[ABC]=1/2 s(1)+1/2 s(2)+1/2 s(3)=1/2 s(1+2+3)=3s
3STEP 3

Area of the whole equilateral triangle

The whole triangle's own area formula gives a second expression.

h=√(s²-s²/4)=√3/2s, [ABC]=√3/4s²
4STEP 4

Set the two areas equal

Setting them equal and solving gives the side.

3s=√3/4s² → 3=√3/4s → s=12/√3
5STEP 5

Simplify the radical

Simplifying the radical gives 4√(3), choice (E).

s=12/√3=12√3/3=4√3
Answer
4√(3)
Check both areas with s=4√3. The pieces give 3s=12√3. The whole gives √3/4s²=√3/4 · 48=12√3 — they agree. Also note the altitude of the whole triangle is h=√3/2 · 4√3=6, exactly the sum 1+2+3 of the three distances; this is Viviani's Theorem, a reassuring cross-check. Numerically 4√3≈6.9, comfortably larger than the biggest perpendicular (3) as any real triangle must be, and it is the only choice matching 12/√3.
💡Key takeaway

Connect the inside point to the corners: the three perpendiculars become heights, and matching the added-up pieces to the whole triangle's area pins down the side.

  • Split the triangle from P
  • Add the three small areas
  • Area of the whole equilateral triangle
  • Set the two areas equal
  • Simplify the radical

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