To find the coordinates of point [tex]\( C \)[/tex], which partitions the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in the ratio [tex]\( 5:8 \)[/tex], we use the section formula for internal division.
Given:
- Coordinates of point [tex]\( A \)[/tex]: [tex]\( (x_1, y_1) = (-2.2, -6.3) \)[/tex]
- Coordinates of point [tex]\( B \)[/tex]: [tex]\( (x_2, y_2) = (2.7, -0.7) \)[/tex]
- Ratio [tex]\( m:n = 5:8 \)[/tex]
First, let's determine the [tex]\( x \)[/tex]-coordinate of point [tex]\( C \)[/tex]:
[tex]\[
x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1
\][/tex]
Substituting the values into the formula:
[tex]\[
x = \left(\frac{5}{5+8}\right)(2.7 - (-2.2)) + (-2.2)
\][/tex]
[tex]\[
x = \left(\frac{5}{13}\right)(2.7 + 2.2) + (-2.2)
\][/tex]
[tex]\[
x = \left(\frac{5}{13}\right)(4.9) + (-2.2)
\][/tex]
[tex]\[
x \approx \left(0.3846\right)(4.9) + (-2.2)
\][/tex]
[tex]\[
x \approx 1.8845 - 2.2
\][/tex]
[tex]\[
x \approx -0.3
\][/tex]
Let's now determine the [tex]\( y \)[/tex]-coordinate of point [tex]\( C \)[/tex]:
[tex]\[
y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1
\][/tex]
Substituting the values into the formula:
[tex]\[
y = \left(\frac{5}{5+8}\right)(-0.7 - (-6.3)) + (-6.3)
\][/tex]
[tex]\[
y = \left(\frac{5}{13}\right)(-0.7 + 6.3) + (-6.3)
\][/tex]
[tex]\[
y = \left(\frac{5}{13}\right)(5.6) + (-6.3)
\][/tex]
[tex]\[
y \approx \left(0.3846\right)(5.6) + (-6.3)
\][/tex]
[tex]\[
y \approx 2.1538 - 6.3
\][/tex]
[tex]\[
y \approx -4.1
\][/tex]
Hence, the coordinates of point [tex]\( C \)[/tex] are:
[tex]\[
(-0.3, -4.1)
\][/tex]
So, the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of point [tex]\( C \)[/tex], to the nearest tenth, are [tex]\(-0.3\)[/tex] and [tex]\(-4.1\)[/tex], respectively.
