To find the coordinates of point [tex]\( C \)[/tex] that partitions the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in the ratio [tex]\( 5:8 \)[/tex], we will use the section formula for internal division.
Given:
- Coordinates of point [tex]\( A \)[/tex]: [tex]\( (-2.2, -6.3) \)[/tex]
- Coordinates of point [tex]\( B \)[/tex]: [tex]\( (-2.4, -6.4) \)[/tex]
- Ratio [tex]\( m:n \)[/tex]: [tex]\( 5:8 \)[/tex]
The coordinates of point [tex]\( C \)[/tex], say [tex]\( (x_C, y_C) \)[/tex], are given by:
[tex]\[ x_C = \left(\frac{m}{m+n}\right) (x_2 - x_1) + x_1 \][/tex]
[tex]\[ y_C = \left(\frac{m}{m+n}\right) (y_2 - y_1) + y_1 \][/tex]
Step-by-step solution:
1. Finding the [tex]\( x \)[/tex]-coordinate of [tex]\( C \)[/tex]:
[tex]\[
x_C = \left(\frac{5}{5+8}\right) (x_2 - x_1) + x_1
\][/tex]
Substitute the values:
[tex]\[
x_C = \left(\frac{5}{13}\right) (-2.4 + 2.2) - 2.2
\][/tex]
Calculate the difference:
[tex]\[
x_C = \left(\frac{5}{13}\right) (-0.2) + (-2.2)
\][/tex]
Multiply:
[tex]\[
x_C = -\left(\frac{1}{13}\right) + (-2.2)
\][/tex]
[tex]\[
x_C = -0.1 + (-2.2) = -2.3
\][/tex]
2. Finding the [tex]\( y \)[/tex]-coordinate of [tex]\( C \)[/tex]:
[tex]\[
y_C = \left(\frac{5}{5+8}\right) (y_2 - y_1) + y_1
\][/tex]
Substitute the values:
[tex]\[
y_C = \left(\frac{5}{13}\right) (-6.4 + 6.3) - 6.3
\][/tex]
Calculate the difference:
[tex]\[
y_C = \left(\frac{5}{13}\right) (-0.1) - 6.3
\][/tex]
Multiply:
[tex]\[
y_C = -\left(\frac{5}{130}\right) + (-6.3)
\][/tex]
[tex]\[
y_C = -0.04 + (-6.3) = -6.3
\][/tex]
After performing all these calculations, we round both coordinates to the nearest tenth if necessary. In this case, rounding is straightforward.
The coordinates of point [tex]\( C \)[/tex] are:
[tex]\[
\boxed{(-2.3, -6.3)}
\][/tex]
