AMC 10 · 2014 · #12
Grade 11 geometry-2dPick an answer.
Nothing in the problem has a size, so Tool #4 (Introduce a Variable) supplies the one length both circles must agree on: the common chord c=AB. Every other quantity gets written in terms of c, and c cancels at the end. Tool #1 (Draw a Diagram) makes the two centers, the two radii and the chord visible, which is what reveals that both centers sit on the perpendicular bisector of AB. Tool #7 (Identify Subproblems) splits the job into one identical subproblem per circle: given a chord and its central angle, find the radius. Tool #3 (Eliminate Possibilities) settles the part the question actually hangs on — the problem asks for larger over smaller, and there are two ways to hand out the arcs, so one of them has to be ruled out by proof rather than by guess.
Find the one shared length
The chord is the one shared length.
The circles meet the rest of the problem only through the chord AB, so that chord is the bridge every comparison must cross.
10.G-CO.A.1Draw A DiagramFold each triangle in half
Folding each triangle halves its angle.
Two radii plus a chord always make an isosceles triangle, and folding it along its axis of symmetry turns it into a right triangle you can do trigonometry in.
Two radii plus a chord always make an isosceles triangle, so folding it along its axis gives a right triangle.
▸ Why?
Every point of a circle sits one radius from the centre, so the two sides from the centre are equal.
▸ Why?
Equal sides make the triangle a mirror image of itself, so the fold line cuts the chord square on at its middle.
Write each radius from the chord
Each radius follows from the chord and a sine.
A chord plus the angle its center views it at pins the radius down completely, so each arc measure secretly names a radius.
10.G-SRT.C.8Introduce A VariableRule out the wrong assignment
The smaller angle belongs to the bigger circle.
Hold a stick of fixed length and walk backwards: the farther away you stand, the narrower the angle it fills, so the narrower 30° view comes from the bigger circle.
10.G-SRT.C.6Eliminate PossibilitiesTurn areas into a radius ratio
The area ratio is the radius ratio squared.
Only the shape of the configuration matters, so every unit of length has to cancel out of the final expression.
7.G.B.4Introduce A VariableGet the exact value of sin 15
The angle addition formula gives an exact value.
An unfamiliar angle becomes familiar the moment you write it as a gap between two angles you already know.
11.F-TF.C.9Identify SubproblemsSimplify to a clean number
Simplifying gives 2+√3, choice (D).
Multiplying by the conjugate turns a stubborn radical denominator into a plain whole number, and the square then reads off in one line.
9.A-SSE.A.2Introduce A VariableThe two circles share one chord, so each radius is fixed by the angle its center views that chord at; the wider 60° view means the smaller circle, and squaring the two views gives 2+√3.
- Find the one shared length
- Fold each triangle in half
- Write each radius from the chord
- Rule out the wrong assignment
- Turn areas into a radius ratio
- Get the exact value of sin 15
- Simplify to a clean number