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13. Find the coordinates of the points of trisection of the line segment joining the points [tex]\((4, -1)\)[/tex] and [tex]\((-2, -3)\)[/tex].

Sagot :

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]