Let's solve the problem of finding the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of point [tex]\( P \)[/tex] on the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] such that [tex]\( P \)[/tex] is [tex]\(\frac{1}{3}\)[/tex] the length of the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex].
The coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are given as [tex]\( A = (0, 3) \)[/tex] and [tex]\( B = (-5, -7) \)[/tex].
Let's set up the formula for finding the coordinates of point [tex]\( P \)[/tex]:
[tex]\[
x = \left(\frac{m}{m+n}\right)\left(x_2 - x_1\right) + x_1
\][/tex]
[tex]\[
y = \left(\frac{m}{m+n}\right)\left(y_2 - y_1\right) + y_1
\][/tex]
Given that [tex]\( P \)[/tex] is [tex]\(\frac{1}{3}\)[/tex] the length of the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], we can set [tex]\( m = 1 \)[/tex] and [tex]\( n = 2 \)[/tex] (since [tex]\( m + n = 3 \)[/tex]).
The coordinates for [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are:
- [tex]\( x_1 = 0 \)[/tex], [tex]\( y_1 = 3 \)[/tex]
- [tex]\( x_2 = -5 \)[/tex], [tex]\( y_2 = -7 \)[/tex]
First, let's find the [tex]\( x \)[/tex]-coordinate of point [tex]\( P \)[/tex]:
[tex]\[
x = \left(\frac{1}{1+2}\right)\left(-5 - 0\right) + 0
\][/tex]
[tex]\[
x = \left(\frac{1}{3}\right)(-5)
\][/tex]
[tex]\[
x = -\frac{5}{3}
\][/tex]
[tex]\[
x \approx -1.6666666666666665
\][/tex]
Next, let's find the [tex]\( y \)[/tex]-coordinate of point [tex]\( P \)[/tex]:
[tex]\[
y = \left(\frac{1}{1+2}\right)\left(-7 - 3\right) + 3
\][/tex]
[tex]\[
y = \left(\frac{1}{3}\right)(-10) + 3
\][/tex]
[tex]\[
y = -\frac{10}{3} + 3
\][/tex]
[tex]\[
y = -\frac{10}{3} + \frac{9}{3}
\][/tex]
[tex]\[
y = -\frac{1}{3}
\][/tex]
[tex]\[
y \approx -0.33333333333333304
\][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are approximately:
[tex]\[
P \left( -1.6666666666666665, -0.33333333333333304 \right)
\][/tex]
