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

Options

A. 12
B. 17
C. 20
D. 25

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 a polygon:

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

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

### Step-by-Step Solution:

1. **Set up the equation using the given number of diagonals:**
We know that the number of diagonals \(D\) is 170. Therefore, we can write the equation:
\[ 170 = \frac{n(n-3)}{2} \]

2. **Multiply both sides by 2 to eliminate the fraction:**
\[ 340 = n(n-3) \]

3. **Rearrange the equation:**
\[ n(n-3) - 340 = 0 \]
This simplifies to:
\[ n^2 - 3n - 340 = 0 \]

4. **Solve the quadratic equation using the factorization method:**
We need to factor the equation \(n^2 - 3n - 340 = 0\). We look for two numbers that multiply to \(-340\) and add to \(-3\). The numbers are \(20\) and \(-17\).

Thus, we can factor the equation as:
\[ (n - 20)(n + 17) = 0 \]

5. **Find the possible values for \(n\):**
Setting each factor to zero gives us:
\[ n - 20 = 0 \quad \Rightarrow \quad n = 20 \]
\[ n + 17 = 0 \quad \Rightarrow \quad n = -17 \]

6. **Determine the valid solution:**
Since the number of sides \(n\) cannot be negative, we discard \(n = -17\). Thus, the number of sides in the polygon is:
\[ n = 20 \]

### Final Answer:
The number of sides in the polygon is **20**.