No. of diagonals of a polygon are 170. No. of sides in this polygon are:

Options

A. 18
B. 20
C. 17
D. None of these

Answer

To find the number of sides in a polygon given that the number of diagonals is 170, we can use the formula for the number of diagonals in an n-sided 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.

### Step 1: Set up the equation
Given that the number of diagonals \(D = 170\), we can set up the equation:

\[ \frac{n(n - 3)}{2} = 170 \]

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

\[ n(n - 3) = 340 \]

### Step 3: Rearrange the equation
Rearranging gives us a standard quadratic equation:

\[ n^2 - 3n - 340 = 0 \]

### Step 4: Factor the quadratic equation
Next, we need to factor the quadratic equation. We look for two numbers that multiply to -340 and add to -3. The numbers are 17 and -20. Thus, we can factor the equation as:

\[ (n - 20)(n + 17) = 0 \]

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

1. \(n - 20 = 0 \Rightarrow n = 20\)
2. \(n + 17 = 0 \Rightarrow n = -17\)

Since the number of sides \(n\) cannot be negative, we discard \(n = -17\).

### Step 6: Conclusion
Thus, the number of sides in the polygon is:

\[ n = 20 \]