Find the co - ordinates of the points of trisection of the line segment joining `(4, -1)` and `(-2, -3)`.

Answer

To find the coordinates of the points of trisection of the line segment joining the points \( A(4, -1) \) and \( B(-2, -3) \), we can follow these steps:

### Step 1: Identify the coordinates of the points
Let the points be:
- \( A(4, -1) \)
- \( B(-2, -3) \)

### Step 2: Calculate the total distance between points A and B
The total distance between the two points can be divided into three equal parts since we are looking for trisection points.

### Step 3: Use the section formula
To find the points of trisection, we can use the section formula. The section formula states that if a point \( P \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) are given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

### Step 4: Find the first point of trisection
Let \( P \) be the first trisection point. Since we want to divide the segment into three equal parts, we can take the ratio \( 1:2 \) (from \( A \) to \( B \)).

Using the section formula:
- \( m = 1 \) (part from A)
- \( n = 2 \) (part from B)

Substituting the coordinates:
\[ P\left(\frac{1 \cdot (-2) + 2 \cdot 4}{1+2}, \frac{1 \cdot (-3) + 2 \cdot (-1)}{1+2}\right) \]

Calculating the x-coordinate:
\[ P_x = \frac{-2 + 8}{3} = \frac{6}{3} = 2 \]

Calculating the y-coordinate:
\[ P_y = \frac{-3 - 2}{3} = \frac{-5}{3} \]

Thus, the coordinates of the first trisection point \( P \) are:
\[ P(2, -\frac{5}{3}) \]

### Step 5: Find the second point of trisection
Let \( Q \) be the second trisection point. For this point, we can take the ratio \( 2:1 \) (from \( A \) to \( B \)).

Using the section formula again:
- \( m = 2 \)
- \( n = 1 \)

Substituting the coordinates:
\[ Q\left(\frac{2 \cdot (-2) + 1 \cdot 4}{2+1}, \frac{2 \cdot (-3) + 1 \cdot (-1)}{2+1}\right) \]

Calculating the x-coordinate:
\[ Q_x = \frac{-4 + 4}{3} = \frac{0}{3} = 0 \]

Calculating the y-coordinate:
\[ Q_y = \frac{-6 - 1}{3} = \frac{-7}{3} \]

Thus, the coordinates of the second trisection point \( Q \) are:
\[ Q(0, -\frac{7}{3}) \]

### Final Answer
The coordinates of the points of trisection of the line segment joining \( (4, -1) \) and \( (-2, -3) \) are:
1. \( P(2, -\frac{5}{3}) \)
2. \( Q(0, -\frac{7}{3}) \)