Given:
The coordinates of the point A and B are (-5,-1) and (4,1)
The point P on the directed line segment from A to B such that P is One-fourth the length of the line segment from A to B.
Thus, we have;
[tex]m=1[/tex] and [tex]m+n=4[/tex]
We need to determine the coordinates of the point P(x,y)
x - coordinates of the point P:
The x - coordinates of the point P can be determined using the formula,
[tex]x=\left(\frac{m}{m+n}\right)\left(x_{2}-x_{1}\right)+x_{1}[/tex]
Substituting the values, we get;
[tex]x=\left(\frac{1}{4}\right)\left(4+5\right)-5[/tex]
[tex]x=\frac{9}{4}-5[/tex]
[tex]x=-\frac{11}{4}[/tex]
Thus, the x - coordinate of the point P is [tex]-\frac{11}{4}[/tex]
y - coordinate of the point P:
The y - coordinate of the point P can be determined using the formula,
[tex]y=\left(\frac{m}{m+n}\right)\left(y_{2}-y_{1}\right)+y_{1}[/tex]
Substituting the values, we get;
[tex]y=\left(\frac{1}{4}\right)\left(1+1\right)-1[/tex]
[tex]y=\frac{2}{4}-1[/tex]
[tex]y=\frac{-2}{4}[/tex]
[tex]y=-\frac{1}{2}[/tex]
Thus, the y - coordinate of the point P is [tex]-\frac{1}{2}[/tex]
Therefore, the coordinates of the point P is [tex]\left(\frac{-11}{4}, \frac{-1}{2}\right)[/tex]
Hence, Option C is the correct answer.
