Find the coordinates of the points which divide the line segment  joining points (-2,2) and (0,8) in four equal parts 

To find the coordinates of the points that divide the line segment joining the points \((-2, 2)\) and \((0, 8)\) into four equal parts, we can follow these steps:

### Step 1: Calculate the total length of the line segment
The line segment is divided into four equal parts, so we will first find the total vector from point \( A(-2, 2) \) to point \( B(0, 8) \).

### Step 2: Determine the vector and intervals
The vector from \( A \) to \( B \) is:

\[
\text{Vector AB} = (x_2 - x_1, y_2 - y_1) = (0 - (-2), 8 - 2) = (2, 6)
\]

Since we need to divide this segment into four equal parts, the ratio for each segment will be \( 1:3, 2:2, \) and \( 3:1 \).

### Step 3: Calculate the coordinates of the dividing points
The points \( P_1 \), \( P_2 \), and \( P_3 \) will divide the line segment into four equal parts.

- **Point \( P_1 \)** divides the line segment in the ratio \( 1:3 \):
  
  The coordinates of \( P_1 \) are given by the section formula:

  \[
  P_1 \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)
  \]

  For \( m = 1 \) and \( n = 3 \):

  \[
  P_1 \left( \frac{1 \cdot 0 + 3 \cdot (-2)}{1+3}, \frac{1 \cdot 8 + 3 \cdot 2}{1+3} \right) = \left( \frac{-6}{4}, \frac{14}{4} \right) = \left( -\frac{3}{2}, \frac{7}{2} \right) = \left( -1.5, 3.5 \right)
  \]

- **Point \( P_2 \)** divides the line segment in the ratio \( 2:2 \):

  \[
  P_2 \left( \frac{2 \cdot 0 + 2 \cdot (-2)}{2+2}, \frac{2 \cdot 8 + 2 \cdot 2}{2+2} \right) = \left( \frac{-4}{4}, \frac{20}{4} \right) = \left( -1, 5 \right)
  \]

- **Point \( P_3 \)** divides the line segment in the ratio \( 3:1 \):

  \[
  P_3 \left( \frac{3 \cdot 0 + 1 \cdot (-2)}{3+1}, \frac{3 \cdot 8 + 1 \cdot 2}{3+1} \right) = \left( \frac{-2}{4}, \frac{26}{4} \right) = \left( -\frac{1}{2}, 6.5 \right) = \left( -0.5, 6.5 \right)
  \]

### Final Answer:
The coordinates of the points that divide the line segment into four equal parts are:

1. \( P_1 \left( -1.5, 3.5 \right) \)
2. \( P_2 \left( -1, 5 \right) \)
3. \( P_3 \left( -0.5, 6.5 \right) \)

Would you like more details or have any further questions?

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Here are 5 related questions to consider:
1. How can you find the ratio in which a line segment is divided by a given point?
2. What are the coordinates of the midpoint of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \)?
3. How do you calculate the area of a triangle formed by three points in the coordinate plane?
4. How can you find the equation of the line passing through two given points?
5. What is the distance between two points in 3-dimensional space?

**Tip:** When dividing a line segment into multiple equal parts, use the section formula with appropriate ratios to find the precise coordinates of each dividing point.

Discover how to find the coordinates of points that divide a line segment into four equal parts using the section formula. This problem involves coordinate geometry and is suitable for high school students.

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