AMC 10 · 2003 · #21

Grade 11 geometry-2d
Archetype Geometric Probability by Area └ Sub-archetype Uniform Draws to Favorable Region
geometric-probabilitylaw-of-cosinespythagorean-identity convert-to-algebraextreme-principle ↑ Prerequisites: trigonometric-ratios
📏 Long solution 💡 3 insights
Problem
An object travels 8 cm straight from A to B. At B it turns by an angle α picked uniformly at random from (0,π) radians, then travels 5 cm straight to C. Find the probability that the straight-line distance AC is less than 7 cm.

Pick an answer.

(A)
$\frac{1}{6}$
(B)
$\frac{1}{5}$
(C)
$\frac{1}{4}$
(D)
$\frac{1}{3}$
(E)
$\frac{1}{2}$
How to solve
Strategy Draw a Diagram

The whole problem hinges on drawing the turn correctly (Tool #1), because "turns at an angle α" and "the angle of the triangle at B" are different angles and mixing them up flips the answer interval. Once the picture is right, put A and B on a coordinate line and write C in terms of α (Tool #4); the distance formula then turns the geometric condition AC ＜ 7 into a plain inequality in cosα (Tool #13). Solving that inequality means finding the one boundary angle where AC equals exactly 7 and seeing which side of it works (Tool #14). Finally, since α is uniform, stop thinking about probability and start measuring length: the answer is the length of the winning set of angles over π (Tool #16).

1STEP 1

Draw the turn, not the triangle angle

The random angle is the turn, so the triangle's angle at B is its supplement.

∠ ABC = π - α
2STEP 2

Put C in coordinates

Coordinates collapse the trigonometry to AC² = 89 + 80cosα.

AC² = (8+5cosα)² + (5sinα)² = 64 + 80cosα + 25(cos²α+sin²α) = 89 + 80cosα
3STEP 3

Turn the condition into an inequality

Comparing squares turns the condition into cosα ＜ -1/2.

89 + 80cosα ＜ 49 ⟹ 80cosα ＜ -40 ⟹ cosα ＜ -1/2
4STEP 4

Find the boundary angle

Cosine falls all the way across the range, so the winners form one interval.

cosα ＜ -1/2 and α ∈ (0,π) ⇔ α ∈ (2π/3, π)
5STEP 5

Measure the winning interval

Uniform means probability is length, so the answer is 1/3, choice (D).

P(AC ＜ 7) = (π - 2π/3)/π = (π/3)/π = 1/3
Answer
1/3
Two independent checks. First, size: at the halfway turn α = π/2 the object goes straight out sideways, giving AC = √(89) ≈ 9.43, comfortably more than 7. So the winning turns all sit in the second half of (0,π), which already forces the probability strictly below 1/2 and rules out (E) before any exact work. Second, robustness against the misreading. Suppose someone reads α as the interior angle ∠ ABC instead of the turn. Then the Law of Cosines gives AC² = 89 - 80cosα ＜ 49, hence cosα ＞ 1/2 and α ∈ (0, π/3) — a different interval, but the same length π/3, so the same probability 1/3. The two readings disagree about which turns work while agreeing on how many, because α ↦ π - α maps one winning set exactly onto the other and does not change length. The answer 1/3 is therefore safe, but the winning interval is (2π/3,π), not (0,π/3): the object has to swing hard back toward where it came from in order to end up close to A.
💡Key takeaway

When an angle is picked evenly at random, probability turns into length: find the stretch of angles that works and see what fraction of the whole range it covers.

  • Draw the turn, not the triangle angle
  • Put C in coordinates
  • Turn the condition into an inequality
  • Find the boundary angle
  • Measure the winning interval

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