AMC 10 · 2021 · #22
Grade 10 geometry-2dPick an answer.
Two circles are never given by their centers here, so chasing angles is slow. But the triangle is right-angled at C, and the two tangent lines are exactly the two legs CB and CA. Put C at the origin with the legs along the axes and the whole picture becomes coordinates. Then each tangency does heavy lifting: a radius must meet its tangent line at a right angle, so O is forced onto the vertical line through B and P onto the horizontal line through A. Each center therefore has only ONE unknown coordinate left. The 'passes through' condition supplies one equation for that one unknown, and the squared term cancels, leaving plain linear algebra. Finding O and finding P are two separate small problems with the same shape, and once both centers are known the distance formula finishes it.
Spot the right angle, set coordinates
The sides form a right triangle.
A 6-8-10 triangle is a doubled 3-4-5, so the right angle is already there, and placing it at the origin turns both tangent lines into the axes.
8.G.B.6Draw A DiagramTangency pins O to one line
Tangency pins the centre to one line.
Touching a line at one point forces the center to sit straight out from that point, along the perpendicular.
Touching a line at one point forces the centre to sit straight out from that point along the perpendicular.
▸ Why?
A tangent meets the radius at the touch point square on, which fixes the direction from the point.
▸ Why?
Every point of the circle sits one radius from the centre, so the distance along that direction is the radius.
Passing through A gives k
Passing through a vertex fixes the position.
Setting two distances from the same center equal always kills the squared term, because the center's own coordinate is squared identically on both sides.
10.G-GPE.A.1Convert To AlgebraRepeat the same two moves for P
The second circle repeats the same two moves.
The second circle is the first one's story with the two legs traded, so the same two moves solve it with no new ideas.
9.A-REI.B.3Identify SubproblemsDistance between the centers
Measuring gives thirty-five twelfths.
Once both centers are ordinary points, the distance is just one use of the Pythagorean theorem on the gap in x and the gap in y.
8.G.B.8Convert To AlgebraWhen a circle touches a line at a named point, its center must sit straight out from that point, so tangency alone knocks a whole unknown out of the problem.
- Spot the right angle, set coordinates
- Tangency pins O to one line
- Passing through A gives k
- Repeat the same two moves for P
- Distance between the centers