AMC 10 · 2023 · #18
Grade 8 geometry-2d
Pick an answer.
Tangency conditions are easy to mis-set-up in words, so Tool #1 (Draw a Diagram) is the first move — place C₁ and C₂ symmetrically on a horizontal axis, mark the centers A, B, and the symmetry axis. Tool #7 (Identify Subproblems) splits the work into two clean pieces: first find C₃'s radius (one tangency equation), then find C₄'s radius (a right triangle plus one tangency equation). Tool #13 (Convert to Algebra) finishes by solving a linear equation in r once the right triangle is set up — the Pythagorean theorem is the workhorse.
Place the two centres
Place them symmetrically.
Grade 6 "place points by their coordinates" — set up the picture so the symmetry is obvious.
6.NS.C.8Draw A DiagramThe third circle's radius
Internal tangency gives its radius.
Grade 7 "facts about circles" — for two internally tangent circles the centers are radius-difference apart.
For two circles touching from the inside, the centres sit a radius difference apart.
▸ Why?
The touching point lies on the line joining the centres, so that line carries both radii.
▸ Why?
Every point of a circle sits one radius from its centre, so those two radii are the only lengths involved.
The fourth circle's internal tangency
Tangency becomes a centre distance.
Grade 8 Pythagorean theorem on the right triangle formed by A, the y-axis, and O₄.
8.G.B.7Draw A DiagramWrite the external tangency
The external tangency is a distance too.
External tangency = "two circles kissing on the outside," so centers are radius-sum apart.
7.G.B.4Identify SubproblemsCombine the equations
Combine the two equations.
Grade 8 "solve a linear equation in one variable" — quadratic terms cancel, leaving a single equation in r.
8.EE.C.7Convert To AlgebraSolve for the radius
Solving gives three twenty-eighths.
One-step linear solve in r — the unique answer drops out.
8.EE.C.7Convert To AlgebraThis AMC 12 problem only needs Grade 8 Pythagorean theorem plus the two simple circle-tangency rules you already know — drop in one right triangle, the squared terms cancel, and r=3/28 falls out of a single linear equation.