AMC 10 · 2011 · #6
Grade 10 geometry-2dPick an answer.
The problem gives no lengths, only a ratio, so everything has to come out of the picture (Tool #1): draw the center O and the radii OB and OC, because the one fact that connects a tangent to the circle's center is that a tangent is perpendicular to the radius at the touch point. Tool #4 turns the ratio 2 : 3 into arc measures 2x and 3x that must sum to 360°. Tool #7 then breaks the figure into two right triangles OBA and OCA and chases angles inside them. Tool #14 handles the question the ratio quietly raises — whether the two tangents meet at all — by testing the boundary case where they would be parallel.
Draw the center and the radii
The centre and two radii complete the picture.
The center is the only point that sees both touch points the same way, so put it in the picture first.
10.G-CO.A.1Draw A DiagramArc length ratio equals arc degree ratio
An arc length ratio is an angle ratio.
Same circle means the same length-per-degree, so a length ratio is a degree ratio in disguise.
On one circle the same length always spans the same number of degrees, so a length ratio is a degree ratio.
▸ Why?
An arc is the share of the whole circle its angle takes, so length and angle are two readings of one thing.
▸ Why?
With the radius fixed the two rise together in step, so their ratio is the same either way you measure.
Split the full turn in ratio 2 to 3
Splitting a full turn gives 144 degrees at the centre.
Five equal parts of 360° are 72° each; two parts and three parts give 144° and 216°.
6.RP.A.3Introduce A VariableCheck the two tangents actually meet
The two tangents really do meet.
Only a diameter makes the two tangents parallel and sends A off to infinity, and 2 : 3 is not a diameter split.
4.G.A.2Extreme PrincipleTangent meets radius at a right angle
Each tangent meets its radius at a right angle.
The shortest way from the center to a tangent line is straight out along the radius, and shortest means perpendicular.
10.G-C.A.2Draw A DiagramThe line to the center bisects the central angle
The line to the centre bisects the central angle.
The figure is a mirror image across line OA, so that line splits the central angle exactly in half.
10.G-SRT.B.5Identify SubproblemsAngle sum in one right triangle
One right triangle gives half the answer, 18 degrees.
In a right triangle the two small angles are complementary, so 90° - 72° = 18°.
8.G.A.5Identify SubproblemsAdd the two halves at A
Doubling gives 36 degrees, choice (C).
Two mirrored 18° slivers meet at A to make 36°.
4.MD.C.7Identify SubproblemsDraw the radii to the two touch points: a tangent always meets its radius at a right angle, and once you know the two arcs are 144° and 216°, the angle between the tangents is just 180° minus the smaller arc, which is 36°.
- Draw the center and the radii
- Arc length ratio equals arc degree ratio
- Split the full turn in ratio 2 to 3
- Check the two tangents actually meet
- Tangent meets radius at a right angle
- The line to the center bisects the central angle
- Angle sum in one right triangle
- Add the two halves at A