A polygon has `90` diagonals , number of its sides is

Options

A. a) `25`
B. b) `17`
C. c) `15`
D. d) `14`

Answer

To find the number of sides of a polygon that has 90 diagonals, we can use the formula for the number of diagonals in a polygon:

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

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

### Step 1: Set up the equation
Given that the number of diagonals \(D = 90\), we can substitute this value into the formula:

\[ 90 = \frac{N(N - 3)}{2} \]

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

\[ 180 = N(N - 3) \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:

\[ N^2 - 3N - 180 = 0 \]

### Step 4: Solve the quadratic equation
Now, we will solve the quadratic equation \(N^2 - 3N - 180 = 0\) using the method of splitting the middle term.

We need to find two numbers that multiply to \(-180\) and add to \(-3\). The numbers \(-15\) and \(12\) fit this requirement:

\[ N^2 - 15N + 12N - 180 = 0 \]

### Step 5: Factor the equation
Now we can factor the equation:

\[ N(N - 15) + 12(N - 15) = 0 \]

This can be factored as:

\[ (N - 15)(N + 12) = 0 \]

### Step 6: Find the possible values for \(N\)
Setting each factor to zero gives us:

1. \(N - 15 = 0 \Rightarrow N = 15\)
2. \(N + 12 = 0 \Rightarrow N = -12\) (not valid since the number of sides cannot be negative)

Thus, the only valid solution is:

\[ N = 15 \]

### Conclusion
The number of sides of the polygon is \(15\).

### Final Answer
Therefore, the correct option is \(C) 15\).

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