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]\(\frac{1}{3}\)[/tex] the length of the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], we can use the section formula for internal division.
Given:
- Coordinates of [tex]\( A \)[/tex] are [tex]\( A(1, 5) \)[/tex]
- Coordinates of [tex]\( B \)[/tex] are [tex]\( B(0, 3) \)[/tex]
- The ratio in which [tex]\( P \)[/tex] divides the line segment [tex]\( AB \)[/tex] is [tex]\( m:n = 1:2 \)[/tex]
The section formula states that if a point [tex]\( P \)[/tex] divides a line segment [tex]\( AB \)[/tex] with coordinates [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], the coordinates of [tex]\( P \)[/tex] are given by:
[tex]\[
P_x = \left( \frac{m}{m+n} \right)(x_2 - x_1) + x_1
\][/tex]
[tex]\[
P_y = \left( \frac{m}{m+n} \right)(y_2 - y_1) + y_1
\][/tex]
Substituting the given values:
- [tex]\( x_1 = 1 \)[/tex]
- [tex]\( y_1 = 5 \)[/tex]
- [tex]\( x_2 = 0 \)[/tex]
- [tex]\( y_2 = 3 \)[/tex]
- [tex]\( m = 1 \)[/tex]
- [tex]\( n = 2 \)[/tex]
First, calculate the [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[
P_x = \left( \frac{1}{1+2} \right)(0 - 1) + 1
\][/tex]
[tex]\[
P_x = \left( \frac{1}{3} \right)(-1) + 1
\][/tex]
[tex]\[
P_x = -\frac{1}{3} + 1
\][/tex]
[tex]\[
P_x = \frac{3}{3} - \frac{1}{3}
\][/tex]
[tex]\[
P_x = \frac{2}{3}
\][/tex]
Next, calculate the [tex]\( y \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[
P_y = \left( \frac{1}{1+2} \right)(3 - 5) + 5
\][/tex]
[tex]\[
P_y = \left( \frac{1}{3} \right)(-2) + 5
\][/tex]
[tex]\[
P_y = -\frac{2}{3} + 5
\][/tex]
[tex]\[
P_y = 5 - \frac{2}{3}
\][/tex]
[tex]\[
P_y = \frac{15}{3} - \frac{2}{3}
\][/tex]
[tex]\[
P_y = \frac{13}{3}
\][/tex]
Thus, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[
P \left( \frac{2}{3}, \frac{13}{3} \right)
\][/tex]
Converted to decimal form, the coordinates are approximately:
[tex]\[
P (0.6666666666666667, 4.333333333333333)
\][/tex]
Therefore, the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of point [tex]\( P \)[/tex] are [tex]\(\boxed{0.6666666666666667, 4.333333333333333}\)[/tex].
