A triangle is formed by choosing 3 vertices a regular hexagon, The probability that it is equilateral triangle is

To solve the problem of finding the probability that a randomly chosen triangle from the vertices of a regular hexagon is equilateral, follow these steps:

### Step 1: Identify the total number of triangles
- A regular hexagon has 6 vertices.
- The number of ways to choose 3 vertices from 6 is given by the combination formula \( \binom{6}{3} \):

\[
\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
\]

So, there are 20 possible triangles.

### Step 2: Identify the number of equilateral triangles
- In a regular hexagon, only specific combinations of vertices can form an equilateral triangle.
- If you label the vertices of the hexagon as \( A_1, A_2, A_3, A_4, A_5, A_6 \), the equilateral triangles that can be formed are:
  1. \( \triangle A_1A_3A_5 \)
  2. \( \triangle A_2A_4A_6 \)

These are the only two equilateral triangles possible.

### Step 3: Calculate the probability
- The probability is the ratio of the number of favorable outcomes (equilateral triangles) to the total number of possible triangles:

\[
\text{Probability} = \frac{\text{Number of Equilateral Triangles}}{\text{Total Number of Triangles}} = \frac{2}{20} = \frac{1}{10}
\]

Thus, the probability that a randomly chosen triangle is equilateral is \( \frac{1}{10} \).

---

Would you like more details or have any questions? Here are 5 related questions:

1. How do you calculate the number of combinations for selecting items?
2. What is a regular polygon, and how does it relate to equilateral triangles?
3. Can you explain the properties of equilateral triangles?
4. How would the problem change if it involved a regular polygon with a different number of sides?
5. What is the probability of selecting an isosceles triangle from the vertices of a regular hexagon?

**Tip:** Remember that equilateral triangles in regular polygons only form when the number of sides is divisible by 3.

Understand the probability of forming an equilateral triangle from the vertices of a regular hexagon. Learn the step-by-step solution, including calculating combinations and identifying equilateral triangle formations. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation