Three vertices out of six vertices of a regular hexagon are chosen at random. The probability that the triangle obtained by joining these vertices is equilateral will be:

Question

Accepted Answer

To determine the probability that a triangle formed by randomly choosing three vertices from a regular hexagon is equilateral, we need to consider the properties of a regular hexagon and the number of ways to choose the vertices.

1. **Total Ways to Choose Three Vertices:**
   A regular hexagon has 6 vertices. The number of ways to choose 3 vertices out of 6 is given by the combination formula:
   $$
   \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
   $$
   So, there are 20 possible ways to choose 3 vertices from the hexagon.

2. **Ways to Form an Equilateral Triangle:**
   In a regular hexagon, an equilateral triangle can be formed by connecting every second vertex. For example, if the vertices are labeled as $ A, B, C, D, E, F $, then the equilateral triangles are $ \triangle ACE $ and $ \triangle BDF $. These are the only two sets of vertices that form equilateral triangles.

3. **Probability Calculation:**
   Since there are 2 equilateral triangles possible out of the 20 ways to choose 3 vertices, the probability that a randomly chosen set of 3 vertices forms an equilateral triangle is:
   $$
   \frac{2}{20} = \frac{1}{10}
   $$

4. **Explanation of Incorrect Options:**
   - **Option 1 ($1/6$):** This would imply that there are more equilateral triangles than actually possible. Specifically, it would suggest 3.33 equilateral triangles, which is not feasible given the structure of a hexagon.
   - **Option 2 ($1/5$):** This would imply that there are 4 equilateral triangles, which is incorrect as only 2 such triangles can be formed.
   - **Option 4 ($1/20$):** This would imply that there is only 1 equilateral triangle, which is incorrect as there are 2 equilateral triangles possible.

Therefore, the correct probability that a randomly chosen set of three vertices forms an equilateral triangle is indeed $\frac{1}{10}$, making Option 3 the correct answer.