To find the coordinates of point [tex]\( P \)[/tex] on the directed line segment from [tex]\( K \)[/tex] to [tex]\( J \)[/tex] such that [tex]\( P \)[/tex] is [tex]\( \frac{3}{5} \)[/tex] the length of the line segment from [tex]\( K \)[/tex] to [tex]\( J \)[/tex], we will use the section formula. The given points are:
- Point [tex]\( K \)[/tex] with coordinates [tex]\((x_1, y_1) = (40, 50)\)[/tex]
- Point [tex]\( J \)[/tex] with coordinates [tex]\((x_2, y_2) = (96, 72)\)[/tex]
We need to find the coordinates of point [tex]\( P \)[/tex], which is [tex]\( \frac{3}{5} \)[/tex] the way from [tex]\( K \)[/tex] to [tex]\( J \)[/tex].
The section formula for point [tex]\( P \)[/tex], which divides the line segment [tex]\( KJ \)[/tex] in the ratio [tex]\( m:n \)[/tex], is given by:
[tex]\[
x = \left( \frac{m}{m + n} \right)(x_2 - x_1) + x_1
\][/tex]
[tex]\[
y = \left( \frac{m}{m + n} \right)(y_2 - y_1) + y_1
\][/tex]
Here, the ratio is [tex]\( m:n = 3:2 \)[/tex] because [tex]\( m = 3 \)[/tex] and [tex]\( m+n = 5 \)[/tex], thus [tex]\( n = 2 \)[/tex].
Let's find the [tex]\( x \)[/tex]-coordinate of point [tex]\( P \)[/tex]:
[tex]\[
x = \left( \frac{3}{3 + 2} \right) (96 - 40) + 40
\][/tex]
Simplify the fraction and the subtraction inside the brackets:
[tex]\[
x = \left( \frac{3}{5} \right) (56) + 40
\][/tex]
Calculate the multiplication:
[tex]\[
x = \left( 0.6 \right) (56) + 40
\][/tex]
[tex]\[
x = 33.6 + 40
\][/tex]
[tex]\[
x = 73.6
\][/tex]
Now, let's find the [tex]\( y \)[/tex]-coordinate of point [tex]\( P \)[/tex]:
[tex]\[
y = \left( \frac{3}{3 + 2} \right) (72 - 50) + 50
\][/tex]
Simplify the fraction and the subtraction inside the brackets:
[tex]\[
y = \left( \frac{3}{5} \right) (22) + 50
\][/tex]
Calculate the multiplication:
[tex]\[
y = \left( 0.6 \right) (22) + 50
\][/tex]
[tex]\[
y = 13.2 + 50
\][/tex]
[tex]\[
y = 63.2
\][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[
(x, y) = (73.6, 63.2)
\][/tex]
