To find the coordinates of point [tex]\( C \)[/tex], which partitions the directed line segment from point [tex]\( A \)[/tex] to point [tex]\( B \)[/tex] in the ratio [tex]\( 5:8 \)[/tex], we will use the section formula.
Given:
- Coordinates of [tex]\( A \)[/tex] are [tex]\( (-2.2, -6.3) \)[/tex]
- Coordinates of [tex]\( B \)[/tex] are [tex]\( (2.7, -0.7) \)[/tex]
- Ratio [tex]\( m:n = 5:8 \)[/tex]
The section formula for the coordinates of a point [tex]\( C \)[/tex] that divides the line segment joining [tex]\( A (x_1, y_1) \)[/tex] and [tex]\( B (x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
\begin{align*}
x &= \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1, \\
y &= \left(\frac{m}{m+n}\right) \left(y_2 - y_1\right) + y_1.
\end{align*}
\][/tex]
First, let's find the [tex]\( x \)[/tex]-coordinate of [tex]\( C \)[/tex]:
[tex]\[
x = \left(\frac{5}{5+8}\right) \left(2.7 - (-2.2)\right) + (-2.2).
\][/tex]
Simplify within the parentheses:
[tex]\[
x = \left(\frac{5}{13}\right) \left(2.7 + 2.2\right) - 2.2.
\][/tex]
Calculate the sum:
[tex]\[
x = \left(\frac{5}{13}\right) \times 4.9 - 2.2.
\][/tex]
Divide [tex]\( 4.9 \)[/tex] by [tex]\( 13 \)[/tex] and then multiply by [tex]\( 5 \)[/tex]:
[tex]\[
x = \left(\frac{5 \times 4.9}{13}\right) - 2.2 = \frac{24.5}{13} - 2.2 = 1.8846 - 2.2.
\][/tex]
Subtract [tex]\( 2.2 \)[/tex]:
[tex]\[
x = -0.3.
\][/tex]
Next, let's find the [tex]\( y \)[/tex]-coordinate of [tex]\( C \)[/tex]:
[tex]\[
y = \left(\frac{5}{5+8}\right) \left(-0.7 - (-6.3)\right) + (-6.3).
\][/tex]
Simplify within the parentheses:
[tex]\[
y = \left(\frac{5}{13}\right) \left(-0.7 + 6.3\right) - 6.3.
\][/tex]
Calculate the sum:
[tex]\[
y = \left(\frac{5}{13}\right) \times 5.6 - 6.3.
\][/tex]
Divide [tex]\( 5.6 \)[/tex] by [tex]\( 13 \)[/tex] and then multiply by [tex]\( 5 \)[/tex]:
[tex]\[
y = \left(\frac{5 \times 5.6}{13}\right) - 6.3 = \frac{28}{13} - 6.3 = 2.1538 - 6.3.
\][/tex]
Subtract [tex]\( 6.3 \)[/tex]:
[tex]\[
y = -4.1.
\][/tex]
Therefore, the coordinates of point [tex]\( C \)[/tex] are [tex]\( (-0.3, -4.1) \)[/tex], rounded to the nearest tenth.
