Number of diagonals in a decagon:

A decagon is a polygon with ten sides. To find the number of diagonals in a decagon, we can use the formula 1– 2 n (n–3), where n represents the number of sides of the polygon.

Applying the formula:

1– 2 n (n–3)

For a decagon, n = 10.

1– 2 * 10 * (10–3)

Simplifying the equation:

1– 2 * 10 * 7

1– 20 * 7

1– 140

Since the formula gives us a negative value, it means that there are no diagonals in a decagon.

Explanation:

The formula 1– 2 n (n–3) represents the number of diagonals in a polygon of n sides. Let's break down this formula to understand it better:

- The term n–3 represents the number of non-adjacent vertices in a polygon. In a decagon, there are 10 vertices, so the number of non-adjacent vertices is 10–3 = 7.

- The term 2 n represents the number of possible diagonals between each non-adjacent vertex. In a decagon, there are 2 * 10 = 20 possible diagonals.

- Finally, the term 1 represents that each diagonal is counted twice, once from each of its endpoints. So, we subtract the total number of diagonals by 1 to avoid double-counting.

In the case of a decagon, the formula gives us a negative value, indicating that there are no diagonals in a decagon. This can be visually confirmed by drawing a decagon and observing that there are no diagonals connecting non-adjacent vertices.

Conclusion:

The number of diagonals in a decagon is none of these options (d) since the formula yields a negative value, indicating that there are no diagonals in a decagon.