AMC 10 · 2018 · #20

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation
area-trianglesequilateral-trianglecoordinate-geometrysymmetry-argument identify-subproblemssymmetry-argumentarea-difference ↑ Prerequisites: area-triangles
📏 Medium solution 💡 3 insights
Problem
A regular hexagon has side length 1. Three points are the midpoints of three alternating sides. The triangle joining alternate corners and the triangle joining the three midpoints overlap, and their overlap is a convex hexagon. Find the area of that overlap.

Pick an answer.

(A)
$\frac {3}{8}\sqrt{3}$
(B)
$\frac {7}{16}\sqrt{3}$
(C)
$\frac {15}{32}\sqrt{3}$
(D)
$\frac {1}{2}\sqrt{3}$
(E)
$\frac {9}{16}\sqrt{3}$
How to solve
Strategy Draw a Diagram

Tool #1 (Draw a Diagram): dropping the figure onto a coordinate grid turns "which points are inside both triangles" into exact equations of lines, so the fuzzy picture becomes numbers. Tool #4 (Introduce a Variable): coordinates let me write each side of each triangle as a line equation, and crossing points come from solving those equations. Tool #7 (Identify Subproblems): the hexagon has 120° rotational symmetry, so the three corners that get sliced off △ ACE are identical — I only have to measure one. Tool #7 (Identify Subproblems): instead of the six-sided overlap directly, I compute the big triangle's area and subtract the three equal corners, two easy pieces instead of one hard one.

1STEP 1

Pin the hexagon to a grid

Pin the hexagon to coordinates.

A(1,0), B(1/2,√3/2), C(-1/2,√3/2), D(-1,0), E(-1/2,-√3/2), F(1/2,-√3/2)
2STEP 2

Find the midpoints; spot two equal triangles

Find the midpoints and draw both triangles.

X(3/4,√3/4), Y(-3/4,√3/4), Z(0,-√3/2); AC=√3
3STEP 3

Cut the overlap out of triangle ACE

Cut three corners out of the big triangle.

[△ ACE]=√3/4(√3)²=3√3/4; overlap=[△ ACE]-3·[corner]
4STEP 4

Measure one corner triangle

By symmetry, measure just one corner.

XZ∩ AC=(5/8,√3/8), XZ∩ AE=(1/4,-√3/4)→[corner]=3√3/32
5STEP 5

Add up the answer

Subtracting gives fifteen root three over thirty-two.

3√3/4-3·3√3/32=24√3/32-9√3/32=15√3/32 → (C)
Answer
15/32√(3)
The overlap must be smaller than each triangle it sits inside. Here [△ ACE]=3√3/4=24√3/32 and [△ XYZ]=√3/4(3/2)²=9√3/16=18√3/32, and our answer 15√3/32 is below both — exactly what a true intersection should do. Writing the choices over 32 gives A=12√3/32, B=14√3/32, C=15√3/32, D=16√3/32, E=18√3/32. Notice E equals [△ XYZ] exactly: that is the trap for anyone who assumes △ XYZ lies fully inside △ ACE — it does not, since its corners poke out. C sits just under D=1/2√3, matching a hexagon that fills most of △ XYZ.
💡Key takeaway

Drop the figure onto coordinates, then build the overlap by cutting three equal corners off the bigger triangle.

  • Pin the hexagon to a grid
  • Find the midpoints; spot two equal triangles
  • Cut the overlap out of triangle ACE
  • Measure one corner triangle
  • Add up the answer

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