Find the co-ordinates of the points fo trisection of the line segment joining the points (-2, 0) and (4, 0).

Answer

To find the coordinates of the points of trisection of the line segment joining the points A(-2, 0) and B(4, 0), we will follow these steps:

### Step 1: Understand the Problem
We need to find two points, P and Q, that divide the line segment AB into three equal parts. This means that the lengths AP, PQ, and QB are equal.

### Step 2: Determine the Ratio of Division
Since P and Q trisect the line segment AB, we can say:
- AP = PQ = QB
- The ratio in which P divides AB is 1:2 (AP:PB).
- The ratio in which Q divides AB is 2:1 (AQ:QB).

### Step 3: Use the Section Formula
The section formula states that if a point divides a line segment joining two points (x1, y1) and (x2, y2) in the ratio m:n, then the coordinates of the point are given by:

\[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

### Step 4: Find Coordinates of Point P
1. **Coordinates of A**: (-2, 0)
2. **Coordinates of B**: (4, 0)
3. **Using the ratio 1:2 for P**:
- \( m = 1 \), \( n = 2 \)
- \( x_P = \frac{1 \cdot 4 + 2 \cdot (-2)}{1 + 2} = \frac{4 - 4}{3} = \frac{0}{3} = 0 \)
- \( y_P = \frac{1 \cdot 0 + 2 \cdot 0}{1 + 2} = \frac{0 + 0}{3} = 0 \)

Thus, the coordinates of point P are (0, 0).

### Step 5: Find Coordinates of Point Q
1. **Using the ratio 2:1 for Q**:
- \( m = 2 \), \( n = 1 \)
- \( x_Q = \frac{2 \cdot 4 + 1 \cdot (-2)}{2 + 1} = \frac{8 - 2}{3} = \frac{6}{3} = 2 \)
- \( y_Q = \frac{2 \cdot 0 + 1 \cdot 0}{2 + 1} = \frac{0 + 0}{3} = 0 \)

Thus, the coordinates of point Q are (2, 0).

### Final Answer
The coordinates of the points of trisection of the line segment joining the points (-2, 0) and (4, 0) are:
- Point P: (0, 0)
- Point Q: (2, 0)