AMC 10 · 2021 · #10
Grade 11 geometry-2dPick an answer.
The phrase "is isosceles" is a three-way condition wearing one name, and the answer choices are built to punish anyone who finds only some of the cases. So Tool #2 (Make a Systematic List) drives everything: name the apex — the vertex where the two equal sides meet — and work through apex A, apex B, apex C as three separate subproblems (Tool #7). Before that casework can be cheap, though, the picture has to be seen: Tool #1 (Draw a Diagram) reveals that (cosθ°, sinθ°) puts all three vertices on one circle of radius 1. That opens the door for Tool #15 (Organize Information in More Ways): on a single circle, side lengths and arc sizes carry exactly the same information, so "two sides are equal" can be rewritten as "two arcs are equal" and the whole problem turns into angle arithmetic instead of distance formulas. Tool #3 (Eliminate Possibilities) does the cleanup at the end, discarding the candidate values of t where C collapses onto a vertex that is already there.
See the circle hiding in the coordinates
All three sit on one circle.
Coordinates written as a cosine and a sine are a circle in disguise, so read them as positions on a dial rather than as pairs of decimals.
11.F-TF.A.2Draw A DiagramTrade side lengths for arc sizes
Trade lengths for arc sizes.
Equal chords on the same circle mean equal arcs, so comparing lengths becomes comparing angles — and angles are far easier to solve for.
Equal chords on the same circle cut equal arcs, so comparing lengths becomes comparing angles.
▸ Why?
Every point of the circle is one radius from the centre, so equal chords make congruent central triangles.
▸ Why?
An arc is the share of the whole circle its central angle takes, so equal angles mean equal arcs.
Name the apex, get three cases
Name the apex to get three cases.
"Isosceles" names a property, not a specific pair of sides, so list every vertex that could be the apex instead of assuming one.
10.G-CO.C.10Make A Systematic ListCase 1: apex at A
The first case gives two values.
Fixing a distance from a point on a circle pins the other point to exactly two places — one on each side.
9.A-REI.B.3Make A Systematic ListCase 2: apex at B
The second gives two more.
The same 20° gap that defines AB can be stepped off from either endpoint, so the second case is the first one mirrored.
9.A-REI.B.3Make A Systematic ListCase 3: apex at C
The third puts it on the perpendicular bisector.
Equidistant from two points means standing on their perpendicular bisector, and on a circle that line runs through the centre, so it marks exactly two spots on the rim.
10.G-CO.C.9Draw A DiagramThrow out the degenerate values
Discard the coinciding values.
A solution that makes two vertices coincide solves the equation but destroys the triangle, so the equation's answers must be filtered by the geometry.
10.G-CO.A.1Eliminate PossibilitiesAdd the surviving values
Adding the survivors gives 380.
The last step answers a different question from the one just solved — the prize is the total, not the list.
4.NBT.B.4Identify SubproblemsWhen points are given as (cosθ, sinθ) they all live on one circle, so equal sides just mean equal arcs — then check every vertex in turn as the apex, because "isosceles" never tells you which two sides match.
- See the circle hiding in the coordinates
- Trade side lengths for arc sizes
- Name the apex, get three cases
- Case 1: apex at A
- Case 2: apex at B
- Case 3: apex at C
- Throw out the degenerate values
- Add the surviving values
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