AMC 10 · 2013 · #12
Grade 11 geometry-2dPick an answer.
"Angles in arithmetic progression" sounds like a condition on three separate numbers, but the 180° angle sum collapses it to a single fact: one angle equals 60°. The collapse has to work in both directions, because we will search for triangles with a 60° angle and then claim they qualify. Once the condition is just one 60° angle, the only freedom left is which side that angle faces. That splits the problem into three law-of-cosines subproblems, one per side, and each is a short piece of algebra.
Angle sum forces a 60 degree angle
The angle sum forces a 60 degree angle.
In an evenly spaced trio the middle term is the average, and the average of a triangle's three angles is always 60°.
In an evenly spaced trio the middle term is the average, and a triangle's three angles always average sixty degrees.
▸ Why?
Terms spaced by the same step balance about the middle, so the middle is exactly the average.
▸ Why?
The three angles always add to a straight angle, so their average is fixed before anything else is known.
Check the reverse direction too
Any step size still gives an even progression.
If one angle is 60°, the other two sit balanced on either side of 60° by themselves, so the spacing is even for free.
10.G-CO.C.10Convert To AlgebraSplit by which side faces 60 degrees
Which side faces it splits the work into three cases.
Naming the side that the known angle faces is the only choice left, and there are just three sides to try.
11.G-SRT.D.11Identify SubproblemsCase: 60 degrees faces x
The first case gives one root.
Two known sides with a known angle between them leave nothing free, so the third side is forced.
11.G-SRT.D.11Convert To AlgebraCase: 60 degrees faces the side 5
The second gives a quadratic.
Fixing the angle between 4 and x still leaves x free, so the constraint arrives as a quadratic instead of a number.
9.A-REI.B.4Convert To AlgebraCase: 60 degrees faces the side 4 fails
The third has no real solution at all.
A negative discriminant is algebra's way of reporting that the picture cannot be drawn at all.
9.A-REI.B.4Eliminate PossibilitiesConfirm both survivors are real triangles
Both survivors are genuine triangles.
Draw the angle, mark off the two lengths, join the ends: a triangle always appears, so a positive root is automatically a real shape.
10.G-SRT.B.5Draw A DiagramAdd them and match the form
Matching the form gives 36, choice (D).
The sum already sits in the requested shape, so the three integers can simply be read off it.
11.N-RN.A.2Introduce A VariableIn a triangle, "the angles are evenly spaced" means nothing more and nothing less than "one angle is 60°" — and then the only question left is which side that angle faces.
- Angle sum forces a 60 degree angle
- Check the reverse direction too
- Split by which side faces 60 degrees
- Case: 60 degrees faces x
- Case: 60 degrees faces the side 5
- Case: 60 degrees faces the side 4 fails
- Confirm both survivors are real triangles
- Add them and match the form
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