To find the coordinates of point [tex]\( P \)[/tex] on the directed line segment from point [tex]\( K \)[/tex] to point [tex]\( J \)[/tex], such that [tex]\( P \)[/tex] is [tex]\(\frac{3}{5}\)[/tex] of the distance from [tex]\( K \)[/tex] to [tex]\( J \)[/tex], we will use the section formula for internal division.
Given:
- Coordinates of [tex]\( K \)[/tex]: [tex]\( K(40, 96) \)[/tex]
- Coordinates of [tex]\( J \)[/tex]: [tex]\( J(85, 105) \)[/tex]
- The ratio in which [tex]\( P \)[/tex] divides the line segment [tex]\( KJ \)[/tex] is [tex]\( \frac{3}{5} \)[/tex], hence [tex]\( \frac{3}{5-3} = \frac{3}{2} \)[/tex].
We apply the section formula:
[tex]\[ x_P = \left( \frac{m}{m+n} \right)(x_2 - x_1) + x_1 \][/tex]
[tex]\[ y_P = \left( \frac{m}{m+n} \right)(y_2 - y_1) + y_1 \][/tex]
Here:
- [tex]\( m = 3 \)[/tex]
- [tex]\( n = 2 \)[/tex] (since [tex]\( m + n = 5 \)[/tex])
- [tex]\( x_1 = 40 \)[/tex]
- [tex]\( y_1 = 96 \)[/tex]
- [tex]\( x_2 = 85 \)[/tex]
- [tex]\( y_2 = 105 \)[/tex]
First, calculate the [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[ x_P = \left( \frac{3}{3+2} \right)(85 - 40) + 40 \][/tex]
[tex]\[ x_P = \left( \frac{3}{5} \right) \times 45 + 40 \][/tex]
[tex]\[ x_P = 0.6 \times 45 + 40 \][/tex]
[tex]\[ x_P = 27 + 40 \][/tex]
[tex]\[ x_P = 67.0 \][/tex]
Next, calculate the [tex]\( y \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[ y_P = \left( \frac{3}{3+2} \right)(105 - 96) + 96 \][/tex]
[tex]\[ y_P = \left( \frac{3}{5} \right) \times 9 + 96 \][/tex]
[tex]\[ y_P = 0.6 \times 9 + 96 \][/tex]
[tex]\[ y_P = 5.4 + 96 \][/tex]
[tex]\[ y_P = 101.4 \][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[ (x_P, y_P) = (67.0, 101.4) \][/tex]
