AMC 10 · 2021 · #2
Grade 6 geometry-2d
Pick an answer.
One corner of the figure points inward, so no single area formula fits it. Tool #7 (Identify Subproblems) is the way in: a compound figure becomes easy the moment it is rewritten as a combination of triangles, and a triangle's area is one multiplication away. Tool #16 (Change Focus / Count the Complement) decides which combination to use — rather than chopping the dart into slivers, it is cleaner to look at the shape that would be there if the dent were filled in, a plain triangle, and then take the filling back out. Tool #1 (Draw a Diagram) does the setup, since the dashed grid already hands over exact coordinates for all four corners and turns the picture into numbers that can be measured.
Read the corners off the grid
Read the corners off the grid.
A grid turns a picture into numbers, so the shape can be measured instead of eyeballed.
5.G.A.1Draw A DiagramFill the dent to get a triangle
Fill the dent to get a triangle.
A shape with a bite taken out is easiest to measure as the whole thing minus the bite.
A shape with a bite taken out is easiest to measure as the whole thing minus the bite.
▸ Why?
The full shape is exactly the region plus the bite, so removing one leaves the other.
▸ Why?
Each piece is a triangle whose area is half its base times its height, both read off the grid.
Measure the big triangle
Measure the big triangle.
With a base flat on an axis, both the base and the height can be counted straight off the coordinates.
6.G.A.1Identify SubproblemsTake out the bite
Taking out the bite gives 6.
Sharing a base means area shrinks in exact step with height, so a height of 2 instead of 5 removes 2/5 of the triangle.
6.G.A.1Identify SubproblemsWhen a shape caves inward, fill the dent to make a plain triangle, measure that, then take the filling back out.
- Read the corners off the grid
- Fill the dent to get a triangle
- Measure the big triangle
- Take out the bite
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