To determine the coordinates of point [tex]\( P \)[/tex] on the directed line segment from [tex]\( K \)[/tex] to [tex]\( J \)[/tex], we can use the section formula in coordinate geometry. The section formula helps to find a point [tex]\( P \)[/tex] that divides the line segment joining two points [tex]\( K \)[/tex] and [tex]\( J \)[/tex] in a given ratio.
Given:
- Coordinates of [tex]\( K \)[/tex]: [tex]\((x_1, y_1) = (40, 96)\)[/tex]
- Coordinates of [tex]\( J \)[/tex]: [tex]\((x_2, y_2) = (85, 105)\)[/tex]
- The ratio in which [tex]\( P \)[/tex] divides the line segment [tex]\( KJ \)[/tex] is [tex]\( \frac{3}{5} \)[/tex].
The section formula states:
[tex]\[
\begin{array}{l}
x = \left( \frac{m}{m + n} \right) \left( x_2 - x_1 \right) + x_1 \\
y = \left( \frac{m}{m + n} \right) \left( y_2 - y_1 \right) + y_1
\end{array}
\][/tex]
where [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are the parts in which the line segment is divided. In this case, [tex]\( P \)[/tex] is [tex]\( \frac{3}{5} \)[/tex] the length from [tex]\( K \)[/tex] to [tex]\( J \)[/tex], so:
- [tex]\( m = 3 \)[/tex]
- The remaining part [tex]\( n = 2 \)[/tex] because [tex]\( \frac{3}{5} \)[/tex] implies there are 5 parts in total and [tex]\( 5 - 3 = 2 \)[/tex].
Substitute [tex]\( x_1 = 40 \)[/tex], [tex]\( x_2 = 85 \)[/tex], [tex]\( y_1 = 96 \)[/tex], [tex]\( y_2 = 105 \)[/tex], [tex]\( m = 3 \)[/tex], and [tex]\( n = 2 \)[/tex] into the formulas:
For [tex]\( x \)[/tex]-coordinate:
[tex]\[
x = \left( \frac{3}{3 + 2} \right) (85 - 40) + 40
\][/tex]
[tex]\[
x = \left( \frac{3}{5} \right) (45) + 40
\][/tex]
[tex]\[
x = 27 + 40
\][/tex]
[tex]\[
x = 67
\][/tex]
For [tex]\( y \)[/tex]-coordinate:
[tex]\[
y = \left( \frac{3}{3 + 2} \right) (105 - 96) + 96
\][/tex]
[tex]\[
y = \left( \frac{3}{5} \right) (9) + 96
\][/tex]
[tex]\[
y = 5.4 + 96
\][/tex]
[tex]\[
y = 101.4
\][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are [tex]\( \boxed{(67.0, 101.4)} \)[/tex].
