Let's determine the number of diagonals in each of these polygons one by one.
### 1) Convex Quadrilateral
A quadrilateral has 4 sides (n = 4). To find the number of diagonals in a polygon, we can use the formula:
[tex]\[ \text{Number of diagonals} = \frac{n(n - 3)}{2} \][/tex]
For a convex quadrilateral:
[tex]\[ n = 4 \][/tex]
[tex]\[ \text{Number of diagonals} = \frac{4(4 - 3)}{2} = \frac{4 \times 1}{2} = \frac{4}{2} = 2 \][/tex]
So, a convex quadrilateral has 2 diagonals.
### 2) Regular Hexagon
A hexagon has 6 sides (n = 6). Using the same formula:
[tex]\[ \text{Number of diagonals} = \frac{n(n - 3)}{2} \][/tex]
For a regular hexagon:
[tex]\[ n = 6 \][/tex]
[tex]\[ \text{Number of diagonals} = \frac{6(6 - 3)}{2} = \frac{6 \times 3}{2} = \frac{18}{2} = 9 \][/tex]
So, a regular hexagon has 9 diagonals.
### 3) Triangle
A triangle has 3 sides (n = 3). Using the same formula:
[tex]\[ \text{Number of diagonals} = \frac{n(n - 3)}{2} \][/tex]
For a triangle:
[tex]\[ n = 3 \][/tex]
[tex]\[ \text{Number of diagonals} = \frac{3(3 - 3)}{2} = \frac{3 \times 0}{2} = \frac{0}{2} = 0 \][/tex]
So, a triangle has 0 diagonals.
### Summary
- A convex quadrilateral has 2 diagonals.
- A regular hexagon has 9 diagonals.
- A triangle has 0 diagonals.
