To determine the [tex]$x$[/tex]- and [tex]$y$[/tex]-coordinates of point [tex]\( C \)[/tex] that partitions the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in the ratio 5:8, we can use the section formula. Let's go through the steps to calculate the coordinates of point [tex]\( C \)[/tex].
Given values:
- Point [tex]\( A = (-2.2, -6.3) \)[/tex]
- Point [tex]\( B = (-2.4, -6.4) \)[/tex]
- Ratio [tex]\( m:n = 5:8 \)[/tex]
The formulas for the coordinates of the point [tex]\( C \)[/tex] that divides the line segment from [tex]\( A(x_1, y_1) \)[/tex] to [tex]\( B(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] are:
[tex]\[
x_C = \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1
\][/tex]
[tex]\[
y_C = \left(\frac{m}{m+n}\right) \left(y_2 - y_1\right) + y_1
\][/tex]
Now, let's plug in the values to find the coordinates of [tex]\( C \)[/tex].
1. Calculate the [tex]$x$[/tex]-coordinate of [tex]\( C \)[/tex]:
[tex]\[
x_1 = -2.2, \quad x_2 = -2.4, \quad m = 5, \quad n = 8
\][/tex]
[tex]\[
x_C = \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1
\][/tex]
[tex]\[
x_C = \left(\frac{5}{5+8}\right) \left(-2.4 - (-2.2)\right) + (-2.2)
\][/tex]
[tex]\[
x_C = \left(\frac{5}{13}\right) \left(-2.4 + 2.2\right) - 2.2
\][/tex]
[tex]\[
x_C = \left(\frac{5}{13}\right) (-0.2) - 2.2
\][/tex]
[tex]\[
x_C = -0.0769 - 2.2
\][/tex]
[tex]\[
x_C \approx -2.3 \quad (\text{rounded to the nearest tenth})
\][/tex]
2. Calculate the [tex]$y$[/tex]-coordinate of [tex]\( C \)[/tex]:
[tex]\[
y_1 = -6.3, \quad y_2 = -6.4, \quad m = 5, \quad n = 8
\][/tex]
[tex]\[
y_C = \left(\frac{m}{m+n}\right) \left(y_2 - y_1\right) + y_1
\][/tex]
[tex]\[
y_C = \left(\frac{5}{5+8}\right) \left(-6.4 - (-6.3)\right) + (-6.3)
\][/tex]
[tex]\[
y_C = \left(\frac{5}{13}\right) \left(-6.4 + 6.3\right) - 6.3
\][/tex]
[tex]\[
y_C = \left(\frac{5}{13}\right) (-0.1) - 6.3
\][/tex]
[tex]\[
y_C = -0.0385 - 6.3
\][/tex]
[tex]\[
y_C \approx -6.3 \quad (\text{rounded to the nearest tenth})
\][/tex]
Therefore, the coordinates of point [tex]\( C \)[/tex] are approximately [tex]\( \boxed{(-2.3, -6.3)} \)[/tex].
