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2013 AMC 10B Problem 12

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2013 AMC 10B Problem 12AMC 10 Step by Step
Name:                             Date: September 13, 2026
Basic Probability · difficulty 2 of 5 · about 2 min · position P11-15

Let SS be the set of sides and diagonals of a regular pentagon. A pair of elements of SS are selected at random without replacement. What is the probability that the two chosen segments have the same length?

  1. A)

    25\frac{2}5

  2. B)

    49\frac{4}9

  3. C)

    12\frac{1}2

  4. D)

    59\frac{5}9

  5. E)

    45\frac{4}5

Answer

B

Key insight

A regular pentagon has 5 equal sides and 5 equal diagonals, so a matching pair is two sides or two diagonals: 2*C(5,2) out of C(10,2).

Solution

A regular pentagon has 55 sides, all of one length, and (52)−5=5\binom{5}{2} - 5 = 5 diagonals, all of another length (by the pentagon's rotational symmetry). So SS has 1010 segments in two length classes of 55 each.

The number of ways to pick two segments is (102)=45\binom{10}{2} = 45. The pair has equal lengths exactly when both are sides or both are diagonals: (52)+(52)=10+10=20\binom{5}{2} + \binom{5}{2} = 10 + 10 = 20 pairs.

The probability is 2045=49\frac{20}{45} = \frac{4}{9}.

The answer is (B) 49\boxed{\textbf{(B)}\ \frac{4}{9}}.

Why this works

Symmetry of the regular pentagon collapses ten segments into just two lengths, turning a geometry question into a two-color counting problem. "Same length" then means "same class," and the count is a sum of within-class pairs.

Alternative approach

Sequentially: whatever the first segment is, 44 of the remaining 99 segments share its length, so the probability is 49\frac{4}{9} directly.

The trap

Forgetting that all five diagonals of a regular pentagon are congruent, or counting ordered pairs in the numerator but unordered in the denominator.

Common mistakes

  • Forgetting that all five diagonals of a regular pentagon are congruent, or counting ordered pairs in the numerator but unordered in the denominator.
  • Using 510\frac{5}{10} for the second pick instead of 49\frac{4}{9}, forgetting the first segment is no longer available.

Techniques

Set up the equation/formula and compute; no special trick needed