Find the coordinates of the points which divide the line segment joining A(-2,2) and (2,8) into four equal parts.

Answer

To find the coordinates of the points that divide the line segment joining A(-2, 2) and B(2, 8) into four equal parts, we will use the midpoint formula. The midpoint formula states that the midpoint M of the line segment joining two points (x1, y1) and (x2, y2) is given by:

\[ M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]

### Step-by-Step Solution:

1. **Identify the Points**:
- Let A = (-2, 2)
- Let B = (2, 8)

2. **Calculate the Midpoint P of A and B**:
- Using the midpoint formula:
\[ P = \left( \frac{-2 + 2}{2}, \frac{2 + 8}{2} \right) \]
- Simplifying:
\[ P = \left( \frac{0}{2}, \frac{10}{2} \right) = (0, 5) \]

3. **Calculate the Midpoint Q of A and P**:
- Using the midpoint formula:
\[ Q = \left( \frac{-2 + 0}{2}, \frac{2 + 5}{2} \right) \]
- Simplifying:
\[ Q = \left( \frac{-2}{2}, \frac{7}{2} \right) = (-1, \frac{7}{2}) \]

4. **Calculate the Midpoint R of P and B**:
- Using the midpoint formula:
\[ R = \left( \frac{0 + 2}{2}, \frac{5 + 8}{2} \right) \]
- Simplifying:
\[ R = \left( \frac{2}{2}, \frac{13}{2} \right) = (1, \frac{13}{2}) \]

5. **Calculate the Midpoint S of Q and R**:
- Using the midpoint formula:
\[ S = \left( \frac{-1 + 1}{2}, \frac{\frac{7}{2} + \frac{13}{2}}{2} \right) \]
- Simplifying:
\[ S = \left( \frac{0}{2}, \frac{20}{4} \right) = (0, 5) \]

### Final Coordinates:
The coordinates of the points that divide the line segment into four equal parts are:
1. Q = (-1, 3.5) or (-1, 7/2)
2. P = (0, 5)
3. R = (1, 6.5) or (1, 13/2)