To find the coordinates of the points of trisection of the line segment joining the points [tex]\((4, -1)\)[/tex] and [tex]\((-2, -3)\)[/tex], follow these steps:
1. Identify the given endpoints:
- Let [tex]\((x_1, y_1) = (4, -1)\)[/tex]
- Let [tex]\((x_2, y_2) = (-2, -3)\)[/tex]
2. Calculate the coordinates of the first trisection point:
- The first trisection point divides the line segment into a ratio of [tex]\(1:2\)[/tex]. This trisection point can be found using the section formula:
[tex]\[
\left(\frac{2x_1 + x_2}{3}, \frac{2y_1 + y_2}{3}\right)
\][/tex]
Plugging in the coordinates:
[tex]\[
\left(\frac{2 \cdot 4 + (-2)}{3}, \frac{2 \cdot (-1) + (-3)}{3}\right) = \left(\frac{8 - 2}{3}, \frac{-2 - 3}{3}\right) = \left(\frac{6}{3}, \frac{-5}{3}\right)
\][/tex]
- The coordinates of the first trisection point are:
[tex]\[
\left(2.0, -1.6666666666666667\right)
\][/tex]
3. Calculate the coordinates of the second trisection point:
- The second trisection point divides the line segment into a ratio of [tex]\(2:1\)[/tex]. This trisection point can be found using the section formula:
[tex]\[
\left(\frac{x_1 + 2x_2}{3}, \frac{y_1 + 2y_2}{3}\right)
\][/tex]
Plugging in the coordinates:
[tex]\[
\left(\frac{4 + 2 \cdot (-2)}{3}, \frac{-1 + 2 \cdot (-3)}{3}\right) = \left(\frac{4 - 4}{3}, \frac{-1 - 6}{3}\right) = \left(\frac{0}{3}, \frac{-7}{3}\right)
\][/tex]
- The coordinates of the second trisection point are:
[tex]\[
\left(0.0, -2.3333333333333335\right)
\][/tex]
Thus, the coordinates of the points of trisection of the line segment joining the points [tex]\((4, -1)\)[/tex] and [tex]\((-2, -3)\)[/tex] are:
[tex]\[
(2.0, -1.6666666666666667) \quad \text{and} \quad (0.0, -2.3333333333333335).
\][/tex]
