AMC 10 · 2016 · #11

Grade 6 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation
area-trianglesarea-differencecoordinate-geometry identify-subproblems ↑ Prerequisites: area-triangles
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
Inside an 8-by-5 rectangle, tick marks cut each side into a piece of length 1 and the rest. A shaded band runs diagonally across the rectangle, touching these marked points. Find the area of the shaded region.

Pick an answer.

(A)
$4\dfrac{3}{5}$
(B)
5
(C)
$5\dfrac{1}{4}$
(D)
$6\dfrac{1}{2}$
(E)
8

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

How to solve
Strategy Identify Subproblems

The shaded band is an awkward six-sided slanted shape, so tool #7 (Identify Subproblems) is the main move: cut it into triangles whose areas are easy and add them up. To get exact lengths, tool #1 (Draw a Diagram) puts the rectangle on a coordinate grid so every corner and mark has a number. Tool #5 (Look for a Pattern) then catches the symmetry: the band looks the same after a half-turn about the rectangle's center, so we only have to measure one half and double it.

1STEP 1

Put the picture on a grid

Placing the rectangle at (0,0),(8,0),(8,5),(0,5), the band's six corners are (7,0),(8,0),(8,1),(0,4),(0,5),(1,5).

vertices: (7,0), (8,0), (8,1), (0,4), (0,5), (1,5)
2STEP 2

Find the center crossing

Both long edges share midpoint (4,2.5), the rectangle's center, so they cross there and pinch the band into two matching halves.

((8+0)/2,(1+4)/2)=(4,2.5), ((1+7)/2,(5+0)/2)=(4,2.5)
3STEP 3

Cut one piece into two triangles

Split the top-left half at center O=(4,2.5) into two triangles, each base 1; heights 4 and 2.5 give areas 2 and 1.25.

1/2· 1· 4=2, 1/2· 1· 2.5=1.25
4STEP 4

Add the halves

The top-left half is 2+1.25=3.25; its half-turn twin matches, so the band is 2×3.25=6.5, i.e. 6 1/2 → (D).

2·(2+1.25)=2· 3.25=6.5=6 1/2 → (D)
Answer
6 1/2
The whole rectangle has area 8× 5=40, and a thin slanted band should be only a small slice of that. Our 6 1/2 is about one-sixth of 40, which fits a narrow strip. It is also one of the listed choices, confirming (D).
💡Key takeaway

Drop the picture on a grid, notice the band pinches at the center, then cut one half into two triangles and double it: 2(2+1.25)=6 1/2.

  • Put the picture on a grid
  • Find the center crossing
  • Cut one piece into two triangles
  • Add the halves

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