AMC 10 · 2017 · #16
Grade 10 geometry-2d
Pick an answer.
Arcs are hard to compute with; centers are easy. The whole problem is five tangency statements, and every one of them says the same kind of thing: the distance between two centers equals a sum or a difference of radii. So tool #1 (Draw a Diagram) is used in a specific way — throw away the arcs and keep only the four centers. Tool #4 (Introduce a Variable) names the unknown radius r and the unknown position of P, which turns "squeezed into the gap" into numbers. Tool #13 (Convert to Algebra) puts coordinates on line JK so each tangency becomes a distance-formula equation. Tool #15 (Organize Information in More Ways) is the finishing move: the three equations all contain x² + y² and r², so subtracting them in pairs deletes every square at once and leaves plain linear equations.
Trade every arc for a distance
Every tangency becomes a centre distance.
Two circles touching is just a fact about how far apart their centers are, so the curves can be forgotten and only the centers kept.
Two circles touching is just a fact about how far apart their centres are.
▸ Why?
The touching point lies on the line joining the centres, so that distance is the radii combined.
▸ Why?
Every point of a circle sits one radius from its centre, so the curves add nothing beyond the radii.
Force out the big radius
Those distances force the outer radius to 3.
The two small diameters, 4 and 2, must exactly tile the big diameter, so JK has no choice but to be 6.
9.A-CED.A.1Introduce A VariableName the unknowns
Coordinates name the remaining unknowns.
Three tangencies give three separate handles on the same unknown point, which is exactly enough to pin down both where P is and how big it is.
10.G-C.A.2Introduce A VariableWrite the three equations
Three tangencies give three equations.
The distance formula is the translator that turns "touches" into an equation you can actually manipulate.
10.G-GPE.A.1Convert To AlgebraSubtract to erase every square
Subtracting erases every square.
When two messy expressions share the same messy part, the difference between them is clean.
9.A-SSE.A.2Organize Information In More WaysSolve and check the circle exists
Solving gives 6/7, choice (B).
One unknown point, three tangency conditions — the system is exactly determined, so the squeezed circle has only one possible size.
9.A-REI.C.6Introduce A VariableWhen circles touch, forget the curves and connect the centers — every tangency becomes a distance you can put in an equation.
- Trade every arc for a distance
- Force out the big radius
- Name the unknowns
- Write the three equations
- Subtract to erase every square
- Solve and check the circle exists