The number of points of intersection of the diagonals of a regular hexagon is :

Options

A. 10
B. 15
C. 18
D. 19

Answer

To find the number of points of intersection of the diagonals of a regular hexagon, we can follow these steps:

### Step 1: Understand the structure of a regular hexagon
A regular hexagon has 6 vertices. Let's label them as A, B, C, D, E, and F.

**Hint:** Remember that a regular hexagon has equal sides and angles, which helps in visualizing the diagonals.

### Step 2: Identify the diagonals
In a hexagon, a diagonal is a line segment connecting two non-adjacent vertices. For each vertex, we can draw diagonals to the other vertices, skipping the adjacent ones.

**Hint:** For each vertex, count how many other vertices can be connected with a diagonal.

### Step 3: Count the total number of diagonals
The formula to calculate the number of diagonals in a polygon with n sides is given by:
\[ \text{Number of Diagonals} = \frac{n(n-3)}{2} \]
For a hexagon (n = 6):
\[ \text{Number of Diagonals} = \frac{6(6-3)}{2} = \frac{6 \times 3}{2} = 9 \]

**Hint:** Use the formula for diagonals to verify your count.

### Step 4: Determine the intersection points of the diagonals
To find the points of intersection of the diagonals, we need to consider that each intersection point is formed by the crossing of two diagonals. For two diagonals to intersect, they must connect four distinct vertices.

**Hint:** Think about how many ways you can choose 4 vertices from the 6 vertices of the hexagon.

### Step 5: Calculate the number of ways to choose 4 vertices
The number of ways to choose 4 vertices from 6 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
For our case:
\[ \binom{6}{4} = \frac{6!}{4! \cdot 2!} = \frac{6 \times 5}{2 \times 1} = 15 \]

**Hint:** Remember that choosing 4 vertices will give you the diagonals that intersect.

### Step 6: Conclusion
Thus, the total number of points of intersection of the diagonals of a regular hexagon is 15.

**Final Answer:** The number of points of intersection of the diagonals of a regular hexagon is 15.