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] into the ratio 5:8, follow these steps:
1. Identify the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Point [tex]\( A \)[/tex] has coordinates [tex]\( (x_1, y_1) = (-22, -6.3) \)[/tex].
- Point [tex]\( B \)[/tex] has coordinates [tex]\( (x_2, y_2) = (-2.4, -6.4) \)[/tex].
2. Identify the ratio [tex]\( m:n \)[/tex]:
- The ratio [tex]\( m:n = 5:8 \)[/tex], where [tex]\( m = 5 \)[/tex] and [tex]\( n = 8 \)[/tex].
3. Calculate the [tex]\( x \)[/tex]-coordinate of point [tex]\( C \)[/tex]:
[tex]\[
x = \left( \frac{m}{m+n} \right) \left( x_2 - x_1 \right) + x_1
\][/tex]
Substituting the values:
[tex]\[
x = \left( \frac{5}{5+8} \right) \left( -2.4 - (-22) \right) + (-22)
\][/tex]
[tex]\[
x = \left( \frac{5}{13} \right) \left( -2.4 + 22 \right) - 22
\][/tex]
[tex]\[
x = \left( \frac{5}{13} \right) \left( 19.6 \right) - 22
\][/tex]
[tex]\[
x = \left( 0.384615 \right) \left( 19.6 \right) - 22
\][/tex]
[tex]\[
x = 7.5384 - 22
\][/tex]
[tex]\[
x \approx -14.5
\][/tex]
4. Calculate the [tex]\( y \)[/tex]-coordinate of point [tex]\( C \)[/tex]:
[tex]\[
y = \left( \frac{m}{m+n} \right) \left( y_2 - y_1 \right) + y_1
\][/tex]
Substituting the values:
[tex]\[
y = \left( \frac{5}{5+8} \right) \left( -6.4 - (-6.3) \right) + (-6.3)
\][/tex]
[tex]\[
y = \left( \frac{5}{13} \right) \left( -6.4 + 6.3 \right) + (-6.3)
\][/tex]
[tex]\[
y = \left( \frac{5}{13} \right) \left( -0.1 \right) + (-6.3)
\][/tex]
[tex]\[
y = \left( 0.384615 \right) \left( -0.1 \right) + (-6.3)
\][/tex]
[tex]\[
y = -0.0385 + (-6.3)
\][/tex]
[tex]\[
y \approx -6.3
\][/tex]
5. State the coordinates of point [tex]\( C \)[/tex]:
- The [tex]\( x \)[/tex]-coordinate of point [tex]\( C \)[/tex] is approximately [tex]\( -14.5 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate of point [tex]\( C \)[/tex] is approximately [tex]\( -6.3 \)[/tex].
Thus, the coordinates of point [tex]\( C \)[/tex], which partitions the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] into the ratio 5:8, are [tex]\( (-14.5, -6.3) \)[/tex] rounded to the nearest tenth.
