AMC 10 · 2016 · #23

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation
area-regular-hexagonequal-spacingratio-proportion convert-to-algebra ↑ Prerequisites: area-regular-hexagon
📏 Long solution 💡 3 insights
Problem
Inside a regular hexagon ABCDEF, four lines run in the same direction and are spaced evenly apart: the side AB, a segment ZW, a segment YX, and the side ED. The points W,X,Y,Z sit on sides BC,CD,EF,FA. These four parallel, equally spaced lines slice the hexagon into three horizontal strips of equal height. Find what fraction of the whole hexagon's area is taken up by the middle hexagon WCXYFZ.

Pick an answer.

(A)
$\frac{1}{3}$
(B)
$\frac{10}{27}$
(C)
$\frac{11}{27}$
(D)
$\frac{4}{9}$
(E)
$\frac{13}{27}$

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

A picture is essential here (tool #1): drawing the hexagon with AB on top and ED on the bottom shows the four parallel lines as horizontal cuts and makes WCXYFZ the middle strip. Tool #7 (Identify Subproblems) breaks the job into clean pieces — total area, where the cutting lines sit, how long ZW and YX are, the area sliced off. Tool #4 (Introduce a Variable) fixes the side length at 1 so every length is a concrete number. The cleanest finish is tool #16 (Count the Complement): instead of measuring the lopsided middle hexagon directly, find the two simple trapezoids trimmed off the top and bottom and subtract them from the whole.

1STEP 1

Set side =1, total area

Only the ratio matters, so set each side to 1. Six equilateral triangles of area √3/4 give the hexagon a total area of 3√3/2.

[ABCDEF]=6·√3/4=3√3/2
2STEP 2

Find the strip height

AB and ED are opposite sides, √3 apart (two apothems), so the three equal strips each have height √3/3.

height=2·√3/2=√3, each strip=√3/3
3STEP 3

Length of ZW and YX

Width grows linearly from 1 at AB to 2 at the middle; ZW sits 2/3 of the way down, so ZW=5/3=YX.

ZW=1+2/3(2-1)=5/3=YX
4STEP 4

Area trimmed off top and bottom

Each outer strip is a trapezoid (bases 1 and 5/3, height √3/3) of area 4√3/9, so the two together total 8√3/9.

2·[1/2(1+5/3)√3/3]=2·4√3/9=8√3/9
5STEP 5

Subtract to get the ratio

Subtract: 3√3/2-8√3/9=11√3/18; dividing by the whole 27√3/18 cancels to give the ratio 11/27, choice (C).

[WCXYFZ]/[ABCDEF]=11√3/18/27√3/18=11/27=(C)
Answer
11/27
The middle strip is the widest of the three (it surrounds the hexagon's fat middle), so it should hold more than one-third of the area but not a huge amount more. The result 11/27≈0.407 is just above 1/3≈0.333, which fits. A quick check on the cut pieces: each trapezoid is 4√3/9=8√3/18, and two of them plus the answer give 16√3/18+11√3/18=27√3/18=3√3/2, the full hexagon — nothing lost. Choice (A) 1/3 is the tempting wrong answer for someone who assumes the three equal-height strips have equal area.
💡Key takeaway

Equal-height strips of a hexagon are not equal-area: cut off the two easy outer trapezoids and the middle slab is left as 11/27 of the whole.

  • Set side =1, total area
  • Find the strip height
  • Length of ZW and YX
  • Area trimmed off top and bottom
  • Subtract to get the ratio

More like this

Same archetype — closest grade level first.