What are the [tex]$x$[/tex]- and [tex]$y$[/tex]-coordinates of point [tex]$E$[/tex], which partitions the directed line segment from [tex]$A$[/tex] to [tex]$B$[/tex] into a ratio of [tex]$1:2$[/tex]?

[tex]\[
\begin{array}{l}
x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1 \\
y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1
\end{array}
\][/tex]

A. [tex]$(0,1)$[/tex]
B. [tex]$(-1,3)$[/tex]
C. [tex]$(-2,5)$[/tex]
D. [tex]$(1,0)$[/tex]



Answer :

To determine the coordinates of point [tex]\( E \)[/tex] that partitions the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in the ratio [tex]\( 1:2 \)[/tex], we follow these steps:

1. Identify the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Point [tex]\( A \)[/tex] has coordinates [tex]\( (0, 1) \)[/tex].
- Point [tex]\( B \)[/tex] has coordinates [tex]\( (-1, 3) \)[/tex].

2. Define the ratio components:
- [tex]\( m = 1 \)[/tex]
- [tex]\( n = 2 \)[/tex]

3. Substitute the coordinates and ratio components into the formula for the [tex]\( x \)[/tex]-coordinate of point [tex]\( E \)[/tex]:
[tex]\[ x_E = \left( \frac{m}{m+n} \right) (x_2 - x_1) + x_1 \][/tex]
Plugging the values in:
[tex]\[ x_E = \left( \frac{1}{1 + 2} \right) ((-1) - 0) + 0 \][/tex]
[tex]\[ x_E = \left( \frac{1}{3} \right) (-1) + 0 \][/tex]
[tex]\[ x_E = -\frac{1}{3} + 0 \][/tex]
[tex]\[ x_E = -\frac{1}{3} \][/tex]

4. Substitute the coordinates and ratio components into the formula for the [tex]\( y \)[/tex]-coordinate of point [tex]\( E \)[/tex]:
[tex]\[ y_E = \left( \frac{m}{m+n} \right) (y_2 - y_1) + y_1 \][/tex]
Plugging the values in:
[tex]\[ y_E = \left( \frac{1}{1 + 2} \right) (3 - 1) + 1 \][/tex]
[tex]\[ y_E = \left( \frac{1}{3} \right) (2) + 1 \][/tex]
[tex]\[ y_E = \frac{2}{3} + 1 \][/tex]
[tex]\[ y_E = \frac{2}{3} + \frac{3}{3} \][/tex]
[tex]\[ y_E = \frac{5}{3} \][/tex]

Therefore, the coordinates of point [tex]\( E \)[/tex] are:
[tex]\[ \left( -\frac{1}{3}, \frac{5}{3} \right) \][/tex]

Simplifying these fractions, we get the coordinates of point [tex]\( E \)[/tex] as:
[tex]\[ (-0.3333, 1.6667) \][/tex]

Thus, the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of point [tex]\( E \)[/tex], which partitions the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in the ratio [tex]\( 1:2 \)[/tex], are approximately [tex]\( -0.3333 \)[/tex] and [tex]\( 1.6667 \)[/tex], respectively.