AMC 10 · 2016 · #8
Grade 6 geometry-2d
Pick an answer.
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.
Put the picture on a grid
A grid names every marked point.
Giving every corner a coordinate turns "how long" and "how far" into plain subtraction.
6.NS.C.8Draw A DiagramFind the center crossing
The two lines cross at the centre.
When both crossing lines have the same midpoint, they cross exactly there and split the figure evenly.
When both crossing segments share the same midpoint, they meet exactly there and split the figure evenly.
▸ Why?
A point that is the middle of a segment lies on that segment's fold line, so a shared middle is a shared point.
▸ Why?
A half turn about that point carries the figure onto itself, so the two halves match exactly.
Cut one piece into two triangles
One half cuts into two easy triangles.
Each shaded sliver is a triangle whose short base is one of the length-1 marks and whose tip reaches the center.
6.G.A.1Identify SubproblemsAdd the halves
Doubling gives 6.5, choice (D).
Measure one half, then double it because the half-turn makes the other half identical.
6.G.A.1Look For A PatternDrop 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
More like this
Same archetype — closest grade level first.
