To find the coordinates of the points of trisection of the line segment joining the points [tex]\((2, -2)\)[/tex] and [tex]\((-1, 4)\)[/tex], we apply the section formula. The section formula helps us to find the coordinates of a point that divides the line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a given ratio [tex]\(m:n\)[/tex].
The coordinates of the point [tex]\(P\)[/tex] dividing the line segment in the ratio [tex]\(m:n\)[/tex] are given by:
[tex]\[ P\left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right) \][/tex]
For trisection, we need to divide the segment into three equal parts, which means we have two points that will divide the segment in the ratio [tex]\(1:2\)[/tex] and [tex]\(2:1\)[/tex].
### First Point of Trisection (ratio [tex]\(1:2\)[/tex])
1. Determine the ratio: [tex]\(1:2\)[/tex]
2. Apply the section formula:
[tex]\[ x_{\text{trisection1}} = \frac{1 \cdot (-1) + 2 \cdot 2}{1 + 2} = \frac{-1 + 4}{3} = \frac{3}{3} = 1 \][/tex]
[tex]\[ y_{\text{trisection1}} = \frac{1 \cdot 4 + 2 \cdot (-2)}{1 + 2} = \frac{4 - 4}{3} = \frac{0}{3} = 0 \][/tex]
So, the coordinates of the first point of trisection are [tex]\((1, 0)\)[/tex].
### Second Point of Trisection (ratio [tex]\(2:1\)[/tex])
1. Determine the ratio: [tex]\(2:1\)[/tex]
2. Apply the section formula:
[tex]\[ x_{\text{trisection2}} = \frac{2 \cdot (-1) + 1 \cdot 2}{2 + 1} = \frac{-2 + 2}{3} = \frac{0}{3} = 0 \][/tex]
[tex]\[ y_{\text{trisection2}} = \frac{2 \cdot 4 + 1 \cdot (-2)}{2 + 1} = \frac{8 - 2}{3} = \frac{6}{3} = 2 \][/tex]
So, the coordinates of the second point of trisection are [tex]\((0, 2)\)[/tex].
### Conclusion
Hence, the coordinates of the points of trisection of the line segment joining [tex]\((2, -2)\)[/tex] and [tex]\((-1, 4)\)[/tex] are:
- First point of trisection: [tex]\((1, 0)\)[/tex]
- Second point of trisection: [tex]\((0, 2)\)[/tex]
