A line segment is of length 5 cm. If the coordinates of its one end are (2,2) and that of the other end are (-1, x), then find the value of x.

Answer

To find the value of \( x \) in the given problem, we will use the distance formula. The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step-by-Step Solution:

1. **Identify the Points**:
- One end of the line segment is \( (2, 2) \) which gives us \( x_1 = 2 \) and \( y_1 = 2 \).
- The other end is \( (-1, x) \) which gives us \( x_2 = -1 \) and \( y_2 = x \).

2. **Set Up the Distance Equation**:
- We know the length of the line segment is 5 cm. Therefore, we can write:
\[ 5 = \sqrt{((-1) - 2)^2 + (x - 2)^2} \]

3. **Square Both Sides**:
- To eliminate the square root, square both sides:
\[ 5^2 = ((-1) - 2)^2 + (x - 2)^2 \]
\[ 25 = (-3)^2 + (x - 2)^2 \]

4. **Calculate the Square**:
- Calculate \((-3)^2\):
\[ 25 = 9 + (x - 2)^2 \]

5. **Isolate the Square Term**:
- Subtract 9 from both sides:
\[ 25 - 9 = (x - 2)^2 \]
\[ 16 = (x - 2)^2 \]

6. **Take the Square Root**:
- Take the square root of both sides:
\[ \sqrt{16} = |x - 2| \]
\[ 4 = |x - 2| \]

7. **Set Up Two Equations**:
- This gives us two cases to solve:
1. \( x - 2 = 4 \)
2. \( x - 2 = -4 \)

8. **Solve for \( x \)**:
- For the first case:
\[ x - 2 = 4 \implies x = 6 \]
- For the second case:
\[ x - 2 = -4 \implies x = -2 \]

9. **Final Values**:
- The possible values for \( x \) are \( 6 \) and \( -2 \).

### Conclusion:
The values of \( x \) that satisfy the conditions of the problem are \( x = 6 \) and \( x = -2 \).