To find the points of trisection of the line segment joining the points [tex]\( A(0, 0) \)[/tex] and [tex]\( B(4, -4) \)[/tex], we need to divide the line segment into three equal parts. Let's denote the two points of trisection as [tex]\( P_1 \)[/tex] and [tex]\( P_2 \)[/tex].
The coordinates of the points of trisection can be found using the section formula. If a point [tex]\( P \)[/tex] divides a line segment joining points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of [tex]\( P \)[/tex] are given by:
[tex]\[
P \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
For points of trisection, we divide the segment [tex]\( AB \)[/tex] in the ratio [tex]\( 1:2 \)[/tex] and [tex]\( 2:1 \)[/tex].
### First Point of Trisection [tex]\( P_1 \)[/tex]
[tex]\( P_1 \)[/tex] divides the segment [tex]\( AB \)[/tex] in the ratio [tex]\( 1:2 \)[/tex]. Using the section formula:
[tex]\[
P_1 \left( \frac{1 \cdot 4 + 2 \cdot 0}{1+2}, \frac{1 \cdot -4 + 2 \cdot 0}{1+2} \right)
\][/tex]
Simplifying:
[tex]\[
P_1 \left( \frac{4}{3}, \frac{-4}{3} \right)
\][/tex]
Thus, the coordinates of [tex]\( P_1 \)[/tex] are:
[tex]\[
P_1 \left( \frac{4}{3}, \frac{-4}{3} \right) \approx (1.333, -1.333)
\][/tex]
### Second Point of Trisection [tex]\( P_2 \)[/tex]
[tex]\( P_2 \)[/tex] divides the segment [tex]\( AB \)[/tex] in the ratio [tex]\( 2:1 \)[/tex]. Using the section formula:
[tex]\[
P_2 \left( \frac{2 \cdot 4 + 1 \cdot 0}{2+1}, \frac{2 \cdot -4 + 1 \cdot 0}{2+1} \right)
\][/tex]
Simplifying:
[tex]\[
P_2 \left( \frac{8}{3}, \frac{-8}{3} \right)
\][/tex]
Thus, the coordinates of [tex]\( P_2 \)[/tex] are:
[tex]\[
P_2 \left( \frac{8}{3}, \frac{-8}{3} \right) \approx (2.667, -2.667)
\][/tex]
So, the points of trisection of the line segment joining the points [tex]\( (0, 0) \)[/tex] and [tex]\( (4, -4) \)[/tex] are approximately:
[tex]\[
\left( 1.333, -1.333 \right) \text{ and } \left( 2.667, -2.667 \right)
\][/tex]
