Find the coordinates of points which divide the line segment joining A (5, - 6) and B (-1, 8) into four equal parts.

Answer

To find the coordinates of the points that divide the line segment joining points A (5, -6) and B (-1, 8) into four equal parts, we will follow these steps:

### Step 1: Identify the points and the ratio
We have two points A (5, -6) and B (-1, 8). We need to divide the line segment AB into four equal parts, which means we will find three points: C, D, and E. The ratios for the segments will be:
- AC:CB = 1:3
- AD:DB = 1:1 (midpoint)
- AE:EB = 3:1

### Step 2: Calculate the coordinates of point C
Using the section formula, the coordinates of a point dividing a line segment in the ratio m:n are given by:
\[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
For point C (1:3):
- \( m = 1 \), \( n = 3 \)
- \( x_1 = 5 \), \( y_1 = -6 \)
- \( x_2 = -1 \), \( y_2 = 8 \)

Calculating the x-coordinate of C:
\[ x_C = \frac{1 \cdot (-1) + 3 \cdot 5}{1 + 3} = \frac{-1 + 15}{4} = \frac{14}{4} = \frac{7}{2} \]

Calculating the y-coordinate of C:
\[ y_C = \frac{1 \cdot 8 + 3 \cdot (-6)}{1 + 3} = \frac{8 - 18}{4} = \frac{-10}{4} = -\frac{5}{2} \]
Thus, the coordinates of point C are \( \left( \frac{7}{2}, -\frac{5}{2} \right) \).

### Step 3: Calculate the coordinates of point D
For point D (1:1):
- \( m = 1 \), \( n = 1 \)

Calculating the x-coordinate of D:
\[ x_D = \frac{1 \cdot (-1) + 1 \cdot 5}{1 + 1} = \frac{-1 + 5}{2} = \frac{4}{2} = 2 \]

Calculating the y-coordinate of D:
\[ y_D = \frac{1 \cdot 8 + 1 \cdot (-6)}{1 + 1} = \frac{8 - 6}{2} = \frac{2}{2} = 1 \]
Thus, the coordinates of point D are \( (2, 1) \).

### Step 4: Calculate the coordinates of point E
For point E (3:1):
- \( m = 3 \), \( n = 1 \)

Calculating the x-coordinate of E:
\[ x_E = \frac{3 \cdot (-1) + 1 \cdot 5}{3 + 1} = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2} \]

Calculating the y-coordinate of E:
\[ y_E = \frac{3 \cdot 8 + 1 \cdot (-6)}{3 + 1} = \frac{24 - 6}{4} = \frac{18}{4} = \frac{9}{2} \]
Thus, the coordinates of point E are \( \left( \frac{1}{2}, \frac{9}{2} \right) \).

### Final Result
The coordinates of points that divide the line segment AB into four equal parts are:
- C: \( \left( \frac{7}{2}, -\frac{5}{2} \right) \)
- D: \( (2, 1) \)
- E: \( \left( \frac{1}{2}, \frac{9}{2} \right) \)