Some points on a plane are marked and they are connected pair wise by line segments . IF the total number of line segments formed is 10 then the number of marked points on the plane is

Answer

To solve the problem, we need to determine the number of marked points on a plane given that the total number of line segments formed is 10. We can use the concept of combinations to find the solution.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that if there are \( x \) points on the plane, the number of line segments (connections) that can be formed by connecting these points pairwise is given by the combination formula \( \binom{x}{2} \).

2. **Setting Up the Equation**:
Since the total number of line segments formed is 10, we can write the equation:
\[ \binom{x}{2} = 10 \]

3. **Using the Combination Formula**:
The combination formula \( \binom{x}{2} \) is calculated as:
\[ \binom{x}{2} = \frac{x!}{2!(x-2)!} = \frac{x(x-1)}{2} \]
Therefore, we can rewrite our equation as:
\[ \frac{x(x-1)}{2} = 10 \]

4. **Eliminating the Fraction**:
To eliminate the fraction, multiply both sides by 2:
\[ x(x-1) = 20 \]

5. **Rearranging the Equation**:
Rearranging gives us a quadratic equation:
\[ x^2 - x - 20 = 0 \]

6. **Factoring the Quadratic**:
We can factor the quadratic equation:
\[ (x - 5)(x + 4) = 0 \]

7. **Finding the Solutions**:
Setting each factor to zero gives us:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]

8. **Considering Valid Solutions**:
Since the number of points cannot be negative, we discard \( x = -4 \). Thus, we have:
\[ x = 5 \]

### Conclusion:
The number of marked points on the plane is **5**.