AMC 10 · 2009 · #22
Grade 10 geometry-3dPick an answer.
Nothing here can be measured until the octahedron is held in a fixed frame, so the first move is to sit it on its three diagonals as coordinate axes. Then the eight faces become eight simple planes, and every plane parallel to a chosen face pair is described by one number. Two subproblems remain: pin that number down (the congruence condition does it, and it is the step most people skip), and then find the polygon. Once the corners of the polygon are located, the side lengths and the area are ordinary plane geometry.
Put the octahedron on coordinate axes
Axes give the solid a clean description.
Standing the octahedron on its three diagonals turns the words "regular octahedron" into one short statement about coordinates.
10.G-GPE.B.4Visualize Spatial RelationshipsWrite the family of parallel cuts
The parallel cuts form a one-parameter family.
The candidate cuts form one sliding stack of parallel planes, so there is exactly one number left to determine.
7.G.A.3Introduce A VariableCongruent pieces force the cut through the center
A point symmetry forces the cut through the centre.
Equal volume rules out every other height, and the point reflection through the center shows the middle cut really does produce congruent pieces.
8.G.A.2Eliminate PossibilitiesLocate where the plane meets the surface
The plane meets the surface at six points.
A quantity that runs from plus s down to minus s along an edge has to pass through zero, and by symmetry it does so at the middle.
7.G.A.3Visualize Spatial RelationshipsEvery side is half an edge
Every resulting side is half an edge.
Each side of the hexagon is a midsegment of one equilateral face, so it is half an edge no matter which face you look at.
Each side of the hexagon is a midsegment of one triangular face, so it is half an edge whichever face you look at.
▸ Why?
Joining two midpoints cuts a triangle in the same ratio on both sides, so the segment is a scaled copy of the base.
▸ Why?
Every face is the same size, so the same argument gives the same length on all of them.
Show the hexagon is regular
Equal distances to the centre make it regular.
Six equal chords, each as long as the radius, have to sit evenly around the circle, and that is exactly what regular means.
8.G.B.8Identify SubproblemsCompute the area of the hexagon
Its area follows from the standard formula.
Spokes from the center chop a regular hexagon into six copies of one equilateral triangle.
6.G.A.1Identify SubproblemsMatch the required form and add
Matching the required form and adding gives 14, choice (E).
The form a root b over c is only pinned down once the fraction is in lowest terms and the radicand is squarefree, so verify both before adding.
6.NS.B.4Eliminate PossibilitiesStand the octahedron on its three diagonals, let the word congruent tell you the cut goes through the center, and the slice becomes a regular hexagon whose side is half an edge.
- Put the octahedron on coordinate axes
- Write the family of parallel cuts
- Congruent pieces force the cut through the center
- Locate where the plane meets the surface
- Every side is half an edge
- Show the hexagon is regular
- Compute the area of the hexagon
- Match the required form and add
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