AMC 10 · 2014 · #19

Grade 8 geometry-2d
Archetype Geometric Probability by Area └ Sub-archetype Uniform Draws to Favorable Region
geometric-probabilitytangent-circlesthirty-sixty-ninety-triangle identify-subproblems ↑ Prerequisites: geometric-probability
📏 Medium solution 💡 3 insights
Problem
Two circles share the same center, with radii 1 and 2. Two points are picked at random, independently and uniformly, on the outer circle of radius 2. Find the probability that the straight segment joining these two points passes through the inner circle of radius 1.

Pick an answer.

(A)
$\frac{1}{6}$
(B)
$\frac{1}{4}$
(C)
$\frac{2-\sqrt{2}}{2}$
(D)
$\frac{1}{3}$
(E)
$\frac{1}{2}$

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

This is a geometry problem about positions and shapes, so tool #1 (Draw a Diagram) carries the work: sketch the two circles and the tangent lines that mark the boundary case. Because everything is symmetric about the center, tool #9 (Solve an Easier Related Problem) lets us freeze the first point wherever we like and only ask where the second point may land. Tool #7 (Identify Subproblems) then splits the job into two clean pieces: first find the boundary angle where a chord just grazes the inner circle, then measure how much of the outer circle gives a chord past that boundary.

1STEP 1

Freeze the first point by symmetry

Spinning the picture about the center changes nothing, so fix the first point A and ask only where the second point B can land.

P = (length of favorable arc for B)/(full circle)
2STEP 2

Draw the tangent lines to find the boundary

The tangent from A touches the inner circle at T with OT perpendicular, so OAT is 30-60-90: hypotenuse 2, leg 1, angle AOT = 60°.

OA = 2, OT = 1, ∠ OTA = 90° → ∠ OAT = 30°, ∠ AOT = 60°
3STEP 3

Turn the tangent into a central-angle rule

Each tangent extended meets the outer circle 120° from A, so AB cuts the inner circle only when the central angle beats that.

chord AB hits inner circle ⇔ ∠ AOB ＞ 120°
4STEP 4

Measure the favorable arc and take the ratio

The favorable arc runs from 120° to 240°, a 120° span out of 360°, so the probability is 1/3 — answer (D).

P = (240° - 120°)/360° = 120°/360° = 1/3 → (D)
Answer
1/3
The favorable arc is exactly one third of the outer circle, so 1/3 is a clean, believable answer that sits sensibly below one half. Sanity check the boundary: a chord tangent to the inner circle is the break-even case, and it corresponds to a 120° central angle, matching the 30-60-90 triangle from the radius and tangent. Everything closer than that misses the inner circle, everything farther cuts it, so a 120°-wide favorable slice giving 1/3 is consistent.
💡Key takeaway

Pin one point down, and the chord only reaches the inner circle when the second point lands more than 120° away, which is a single one-third slice of the outer circle.

  • Freeze the first point by symmetry
  • Draw the tangent lines to find the boundary
  • Turn the tangent into a central-angle rule
  • Measure the favorable arc and take the ratio

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