Find the points of trisectrion of the linear segment joining the points (1, -2) and (-3, 4) .

Answer

To find the points of trisection of the line segment joining the points (1, -2) and (-3, 4), we will follow these steps:

### Step 1: Understand the concept of trisection
Trisection means dividing the line segment into three equal parts. Therefore, we need to find two points that divide the segment into three equal lengths.

### Step 2: Identify the endpoints
The endpoints of the line segment are given as:
- Point A (x1, y1) = (1, -2)
- Point B (x2, y2) = (-3, 4)

### Step 3: Use the section formula
The section formula states that if a point P divides the line segment joining points (x1, y1) and (x2, y2) 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 trisection point (Point P)
Let’s denote the first trisection point as P, which divides the segment in the ratio 1:2 (1 part from A and 2 parts from B).

Using the section formula:
- m = 1, n = 2
- x1 = 1, y1 = -2
- x2 = -3, y2 = 4

Calculating the x-coordinate of P:
\[ x_P = \frac{1 \cdot (-3) + 2 \cdot 1}{1 + 2} = \frac{-3 + 2}{3} = \frac{-1}{3} \]

Calculating the y-coordinate of P:
\[ y_P = \frac{1 \cdot 4 + 2 \cdot (-2)}{1 + 2} = \frac{4 - 4}{3} = \frac{0}{3} = 0 \]

Thus, the coordinates of point P are:
\[ P\left(-\frac{1}{3}, 0\right) \]

### Step 5: Find the second trisection point (Point Q)
Let’s denote the second trisection point as Q, which divides the segment in the ratio 2:1 (2 parts from A and 1 part from B).

Using the section formula:
- m = 2, n = 1

Calculating the x-coordinate of Q:
\[ x_Q = \frac{2 \cdot (-3) + 1 \cdot 1}{2 + 1} = \frac{-6 + 1}{3} = \frac{-5}{3} \]

Calculating the y-coordinate of Q:
\[ y_Q = \frac{2 \cdot 4 + 1 \cdot (-2)}{2 + 1} = \frac{8 - 2}{3} = \frac{6}{3} = 2 \]

Thus, the coordinates of point Q are:
\[ Q\left(-\frac{5}{3}, 2\right) \]

### Final Result
The points of trisection of the line segment joining the points (1, -2) and (-3, 4) are:
- Point P: \(\left(-\frac{1}{3}, 0\right)\)
- Point Q: \(\left(-\frac{5}{3}, 2\right)\)