Question

Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral equals:

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{5} \)
• C. \( \frac{1}{10} \)
• D. \( \frac{1}{20} \)

Hint:

When selecting vertices of a regular polygon, consider symmetry to count the number of possible equilateral triangles.

Answer 1

The Correct Option is C

A regular hexagon has 6 vertices. The number of ways to choose 3 vertices from 6 is given by: \[ \binom{6}{3} = 20 \] Step 1: Count the number of equilateral triangles In a regular hexagon, there are exactly 2 ways to select 3 vertices that form an equilateral triangle (they must be spaced 120 degrees apart). Step 2: Calculate the probability The probability of selecting an equilateral triangle is: \[ \frac{2}{20} = \frac{1}{10} \] Thus, the correct answer is \( \frac{1}{10} \).