Sure, let’s determine 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 segment from [tex]\( K \)[/tex] to [tex]\( J \)[/tex]. The coordinates for points [tex]\( K \)[/tex] and [tex]\( J \)[/tex] are given as follows:
- [tex]\( K = (40, 96) \)[/tex]
- [tex]\( J = (85, 105) \)[/tex]
We need to use the given formula for the coordinates of point [tex]\( P \)[/tex]:
[tex]\[ x = \left( \frac{m}{m+n} \right) \left( x_2 - x_1 \right) + x_1 \][/tex]
[tex]\[ y = \left( \frac{m}{m+n} \right) \left( y_2 - y_1 \right) + y_1 \][/tex]
The ratio given is [tex]\(\frac{3}{5}\)[/tex], where:
[tex]\[ m = 3 \][/tex]
[tex]\[ n = 2 \][/tex] (since in the ratio [tex]\(\frac{m}{m+n} = \frac{3}{5}\)[/tex], we have [tex]\(m+n = 5\)[/tex], thus [tex]\(n = 5 - 3 = 2\)[/tex])
Now, let’s plug these values into the formulas for the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of [tex]\( P \)[/tex].
### Calculating the x-coordinate:
[tex]\[ x_P = \left( \frac{3}{3+2} \right) \left( 85 - 40 \right) + 40 \][/tex]
[tex]\[ x_P = \left( \frac{3}{5} \right) \left(45 \right) + 40 \][/tex]
[tex]\[ x_P = 27 + 40 \][/tex]
[tex]\[ x_P = 67 \][/tex]
### Calculating the y-coordinate:
[tex]\[ y_P = \left( \frac{3}{3+2} \right) \left( 105 - 96 \right) + 96 \][/tex]
[tex]\[ y_P = \left( \frac{3}{5} \right) \left( 9 \right) + 96 \][/tex]
[tex]\[ y_P = 5.4 + 96 \][/tex]
[tex]\[ y_P = 101.4 \][/tex]
Thus, the coordinates of point [tex]\( P \)[/tex] are [tex]\((67.0, 101.4)\)[/tex].
