AMC 10 · 2016 · #17
Grade 10 geometry-2dPick an answer.
The phrase "center of an equilateral triangle" is a position, not a number, so tool #1 (Draw a Diagram) with coordinate axes is the move that makes it computable: put the square on the grid and every center becomes an ordered pair. Tool #9 (Solve an Easier Related Problem) fixes the side length at 1, which is free because the question asks for a ratio and scaling the whole picture scales both areas equally. Tool #7 (Identify Subproblems) splits the work into three small, independent questions — how tall is an equilateral triangle, where inside it is the center, and how do you get the area of a square from its diagonal. Tool #17 (Visualize Spatial Relationships) supplies the fact that a quarter turn about the square's center carries the whole picture onto itself, which both proves EFGH is a square and hands over its center for free. Tool #4 (Introduce a Variable) names the half-diagonal R so the final area is one clean formula, and tool #3 (Eliminate Possibilities) confirms the closed form against the five decimal values at the end.
Fix the side length at 1
A ratio lets the side be fixed at one.
A ratio does not care about the size of the picture, so pick the size that makes the arithmetic easiest.
7.G.A.1Solve An Easier Related ProblemPut the square on the grid
A grid places every corner.
Coordinates turn the words "outside the square" into a direction you can add to a midpoint.
6.G.A.3Draw A DiagramHeight of an equilateral triangle
The triangle's height follows at once.
Slicing an equilateral triangle down the middle leaves a right triangle whose missing leg is exactly what Pythagoras returns.
8.G.B.7Identify SubproblemsLocate the center E
That locates one centre.
The center is the balance point, and it hangs one third of the way up from whichever side you call the base.
10.G-CO.C.10Identify SubproblemsSpin the picture a quarter turn
A quarter turn gives the other three.
The whole construction is built the same way on all four sides, so a quarter turn cannot tell the difference — and that symmetry is what forces EFGH to be a square.
The construction is built the same way on all four sides, so a quarter turn cannot tell the difference.
▸ Why?
The turn moves the figure onto itself without stretching, so every length is repeated exactly.
▸ Why?
Four quarter turns fill the whole turn about the centre, so the four sides account for everything.
Area from the half-diagonal
The half-diagonal gives the area.
Once you know how far a square's corner is from its center, the whole square is determined.
10.G-GPE.B.7Introduce A VariableCross-check and take the ratio
A side check confirms choice (B).
Two different routes to the same area — one through the diagonal, one through the side — is the cheapest way to catch an arithmetic slip.
10.G-GPE.B.4Eliminate PossibilitiesThe center of an equilateral triangle sits one third of the way up from its base, so each corner of EFGH pokes out √(3)/6 past a side of the unit square — and that alone fixes the area ratio at (2+√(3))/3.
- Fix the side length at 1
- Put the square on the grid
- Height of an equilateral triangle
- Locate the center E
- Spin the picture a quarter turn
- Area from the half-diagonal
- Cross-check and take the ratio