To find the coordinates of point [tex]\( P \)[/tex] on the directed line segment from point [tex]\( A \)[/tex] to point [tex]\( B \)[/tex], such that [tex]\( P \)[/tex] is [tex]\(\frac{1}{4} \)[/tex] the length of the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], you can use the section formula for a given ratio.
The section formula to find the coordinates of a point that divides a line segment joining two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
(x, y) = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)
\][/tex]
Here, the points are:
- [tex]\( A = \left(\frac{-29}{4}, \frac{-3}{2}\right) \)[/tex]
- [tex]\( B = \left(\frac{25}{4}, \frac{-1}{2}\right) \)[/tex]
- The ratio in which point [tex]\( P \)[/tex] divides the segment [tex]\( AB \)[/tex] is [tex]\(\frac{1}{4} \)[/tex], so [tex]\( m = 1 \)[/tex] and [tex]\( n = 4 - 1 = 3 \)[/tex].
Using the section formula, plug in the values:
[tex]\[
P_x = \frac{1 \cdot \frac{25}{4} + 3 \cdot \frac{-29}{4}}{1 + 3}
\][/tex]
[tex]\[
P_x = \frac{\frac{25}{4} + \frac{-87}{4}}{4}
\][/tex]
[tex]\[
P_x = \frac{\frac{25 - 87}{4}}{4}
\][/tex]
[tex]\[
P_x = \frac{\frac{-62}{4}}{4}
\][/tex]
[tex]\[
P_x = \frac{-62}{16}
\][/tex]
[tex]\[
P_x = -\frac{31}{8}
\][/tex]
[tex]\[
P_x = -3.875
\][/tex]
Now for the [tex]\( y \)[/tex]-coordinate:
[tex]\[
P_y = \frac{1 \cdot \left(\frac{-1}{2}\right) + 3 \cdot \left(\frac{-3}{2}\right)}{1 + 3}
\][/tex]
[tex]\[
P_y = \frac{\frac{-1}{2} + \frac{-9}{2}}{4}
\][/tex]
[tex]\[
P_y = \frac{\frac{-10}{2}}{4}
\][/tex]
[tex]\[
P_y = \frac{-10}{8}
\][/tex]
[tex]\[
P_y = -1.25
\][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are [tex]\( \left( -3.875, -1.25 \right) \)[/tex].
So, the coordinates of point [tex]\( P \)[/tex] are [tex]\( \left( -3.875, -1.25 \right) \)[/tex].
