AMC 10 · 2006 · #17
Grade 10 geometry-2d
Pick an answer.
The picture hands you exactly one measurement, AF, and asks for a ratio of two unknowns. One equation, two unknowns — so the finish cannot be ordinary algebra alone. Tool #13 (Convert to Algebra) turns the geometry into that one equation: tools #17 and #1 pin down where E actually sits and put the square on axes, tool #7 splits off the tangent-length subproblem AF² = AE² - r², and tool #4 names the coordinates. What finishes the problem is tool #15 (Organize Information in More Ways): the equation lands in the shape s² + √(2) sr = 9 + 5√(2), and re-reading it as "rational part" plus "√(2) part" — which is legal only because r and s are rational — splits one equation into two. That split is the whole problem; everything before it is bookkeeping.
Pin down where E sits
Two clues force the centre onto the diagonal, past the corner.
Every point of a circle is one radius from the center, so "the circle passes through D" is really just the length fact ED = r.
10.G-C.A.2Visualize Spatial RelationshipsPut the square on axes
Coordinates write the centre with one expression.
Sliding a distance r along a diagonal adds r/√(2) to each coordinate, because the diagonal splits that step evenly between the two directions.
10.G-GPE.B.4Draw A DiagramTrade the tangent for a right angle
The tangent becomes a right angle, so the touch point never needs locating.
A tangent length is just a leg of a right triangle, so knowing AF is the same as knowing how far A is from the center.
8.G.B.7Identify SubproblemsCompute AE squared
Computing the long distance makes the squared radius cancel.
The two legs each carry half of r², so the radius rebuilds itself and then cancels, leaving a clean statement about s and r only.
10.G-GPE.B.7Convert To AlgebraSplit the equation at the radical
Rationality splits the equation at the radical into two equations.
A rational number can never equal a nonzero rational times √(2), so the plain part and the √(2) part of an equation have to match up separately.
A rational number can never equal a nonzero rational times root two, so the plain part and the root part must match separately.
▸ Why?
Two is not a perfect square in any prime recipe, so its square root cannot be written as a ratio of whole numbers.
▸ Why?
When an identity holds, each independent kind of term on one side must match the same kind on the other.
Solve and take the ratio
Solving gives the ratio 5/9, choice (B).
Once the equation splits in two, each piece names one unknown, and dividing them gives the ratio directly.
9.A-REI.B.4Introduce A VariableThe tangent turns into a right triangle and the whole picture collapses to s² + √(2) sr = 9 + 5√(2); because r and s are rational, the plain parts and the √(2) parts must match separately, so s²=9 and sr=5 and r/s = 5/9.
- Pin down where E sits
- Put the square on axes
- Trade the tangent for a right angle
- Compute AE squared
- Split the equation at the radical
- Solve and take the ratio