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

Possible coordinates:
A. (0, 1)
B. (1, 0)
C. [tex][tex]$(-2,5)$[/tex][/tex]



Answer :

GinnyAnswer
To determine the coordinates of point [tex]\( E \)[/tex] that partitions the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] into a ratio of [tex]\( 1:2 \)[/tex], we can use the section formula. The section formula for a point that divides a line segment in the ratio [tex]\( m:n \)[/tex] is given by:

[tex]\[ (x, y) = \left(\frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n}\right) \][/tex]

Given:
- [tex]\( A = (0, 1) \)[/tex]
- [tex]\( B = (-2, 5) \)[/tex]
- The ratio [tex]\( m:n = 1:2 \)[/tex]
- [tex]\( m = 1 \)[/tex]
- [tex]\( n = 2 \)[/tex]

Let's apply these to the section formula step by step.

Step 1: Calculate the x-coordinate of point [tex]\( E \)[/tex]:

[tex]\[ x_E = \frac{m \cdot x_B + n \cdot x_A}{m+n} = \frac{1 \cdot (-2) + 2 \cdot (0)}{1 + 2} = \frac{-2 + 0}{3} = \frac{-2}{3} = -\frac{2}{3} \][/tex]

Step 2: Calculate the y-coordinate of point [tex]\( E \)[/tex]:

[tex]\[ y_E = \frac{m \cdot y_B + n \cdot y_A}{m+n} = \frac{1 \cdot 5 + 2 \cdot 1}{1 + 2} = \frac{5 + 2}{3} = \frac{7}{3} \][/tex]

Thus, the coordinates of point [tex]\( E \)[/tex] are:

[tex]\[ E = \left( -\frac{2}{3}, \frac{7}{3} \right) \][/tex]

For convenience and better understanding, we can convert these to decimal form:

[tex]\[ E = (-0.6666666666666666, 2.333333333333333) \][/tex]

In conclusion, the [tex]\( x \)[/tex]-coordinate is [tex]\( -0.6666666666666666 \)[/tex] and the [tex]\( y \)[/tex]-coordinate is [tex]\( 2.333333333333333 \)[/tex]. Therefore, point [tex]\( E \)[/tex] that divides the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in the ratio [tex]\( 1:2 \)[/tex] has coordinates:

[tex]\[ E = (-0.6666666666666666, 2.333333333333333) \][/tex]

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