AMC 10 · 2022 · #25
Grade 10 geometry-2dnumber-theoryPick an answer.
Tool #1 (Diagram): the segment plus the two axis pieces closes a right triangle with legs a, b and hypotenuse c, and the circle is tangent to all three sides. Tool #15 (Organize in More Ways): there are exactly two ways a circle can be tangent to all three sides of that triangle — as the incircle, or as the excircle opposite the right angle — and both must be counted. Tool #13 (Convert to Algebra) is the crux: each configuration gives a linear relation between a, b, c, r, and feeding it into c² = a² + b² collapses both cases into the single factored equation (a - 2r)(b - 2r) = 2r². After that, tool #2 (Systematic List) turns "count the segments" into "count the divisor pairs of 2r²", tool #6 (Guess and Check) walks r = 1, 2, 3, … until the count reaches 14, and tool #3 (Eliminate) checks the final ratio against the five choices.
Close it into a right triangle
Close the segment into a right triangle.
The circle already touches both axes for free, so the only new condition is that it touches the slanted side too.
10.G-GPE.B.4Draw A DiagramTwo ways the circle can fit
The circle sits inside or outside.
Missing the excircle case is the whole trap: the smallest segment in the final list comes from that family, not the inscribed one.
10.G-C.A.3Organize Information In More WaysFeed it into Pythagoras
Both cases give the same equation.
Squaring kills the sign difference between the two cases, so the two pictures merge into one equation.
9.A-CED.A.2Convert To AlgebraUse the factoring trick
Factoring turns it into a divisor problem.
Adding the missing corner term turns a scattered four-term expression into one clean product of two factors.
Adding the missing corner term turns a scattered expression into one clean product of two factors.
▸ Why?
Expanding a product of two brackets spreads every term across the other, which the rewrite reverses.
▸ Why?
Once it is a product with a fixed value, the possibilities are exactly its factor pairs.
Count the first family
Count the case with both factors positive.
Counting tangent lines has quietly turned into counting divisors.
4.OA.B.4Make A Systematic ListCount the second family
Count the case with both negative.
The negative branch survives only in the narrow band r ＜ u ＜ 2r, which is why it is easy to overlook.
7.EE.B.4Make A Systematic ListWalk the radius upward
Increase it until the count reaches fourteen.
72 is the first 2r² rich enough in divisors, because 6 is the first radius whose square carries three different prime powers.
4.OA.B.4Guess And CheckFind the extreme lengths
Find the shortest and the longest.
The tiny 3-4-5 triangle and the sprawling 13-84-85 triangle share the very same circle of radius 6.
8.G.B.7Make A Systematic ListTake the ratio
The ratio is 17.
Both extremes are hypotenuses of right triangles sharing one circle, so their ratio is just 85 divided by 5.
6.RP.A.1Eliminate PossibilitiesA circle centered at (r, r) with radius r hugs both axes, so each tangent segment closes a right triangle around it — either inside as the incircle or outside as the excircle at the right angle. Both cases give the same equation (a - 2r)(b - 2r) = 2r², so counting segments is really counting divisors of 2r²: the first radius with 14 of them is r = 6, and 85/5 = 17.
- Close the segment into a right triangle
- Two ways the circle can fit
- Feed each case into Pythagoras
- Factor with Simon's trick
- Family one: both factors positive
- Family two: both factors negative
- Walk r upward until the count hits 14
- List the 14 segments at r = 6
- Take the ratio