A convex polygon has 65 diagonals. Find number of sides of polygon.

Answer

To find the number of sides of a convex polygon that has 65 diagonals, we can use the formula for the number of diagonals in a polygon, which is given by:

\[ D = \frac{n(n-3)}{2} \]

where \( D \) is the number of diagonals and \( n \) is the number of sides of the polygon.

### Step 1: Set up the equation
Since we know that the polygon has 65 diagonals, we can set up the equation:

\[ \frac{n(n-3)}{2} = 65 \]

### Step 2: Eliminate the fraction
To eliminate the fraction, multiply both sides of the equation by 2:

\[ n(n-3) = 130 \]

### Step 3: Rearrange the equation
Now, rearranging the equation gives us:

\[ n^2 - 3n - 130 = 0 \]

### Step 4: Factor the quadratic equation
Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to -130 and add to -3. The numbers -13 and 10 fit this requirement:

\[ (n - 13)(n + 10) = 0 \]

### Step 5: Solve for \( n \)
Setting each factor equal to zero gives us:

1. \( n - 13 = 0 \) → \( n = 13 \)
2. \( n + 10 = 0 \) → \( n = -10 \) (not valid since the number of sides cannot be negative)

Thus, the only valid solution is:

\[ n = 13 \]

### Conclusion
The number of sides of the polygon is **13**.

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