Given two points A (-5, 2) and B (1, -4), find :
mid-point of AB.

Answer

To find the mid-point of the line segment joining the points A (-5, 2) and B (1, -4), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coordinates of Points A and B:**
- Point A has coordinates \( A(-5, 2) \) where \( x_1 = -5 \) and \( y_1 = 2 \).
- Point B has coordinates \( B(1, -4) \) where \( x_2 = 1 \) and \( y_2 = -4 \).

2. **Use the Mid-point Formula:**
The mid-point \( M \) of the line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

3. **Substitute the Values into the Formula:**
- For the x-coordinate:
\[ M_x = \frac{x_1 + x_2}{2} = \frac{-5 + 1}{2} \]
- For the y-coordinate:
\[ M_y = \frac{y_1 + y_2}{2} = \frac{2 + (-4)}{2} \]

4. **Calculate the x-coordinate:**
\[ M_x = \frac{-5 + 1}{2} = \frac{-4}{2} = -2 \]

5. **Calculate the y-coordinate:**
\[ M_y = \frac{2 - 4}{2} = \frac{-2}{2} = -1 \]

6. **Combine the Coordinates to Get the Mid-point:**
Therefore, the mid-point \( M \) is:
\[ M = (-2, -1) \]

### Final Answer:
The mid-point of the line segment AB is \( M(-2, -1) \).

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