The coordinates of the points A and B are not given. We'll assume: A(3,1) B(12,-5)
Answer:
The coordinates of E are (6,-1)
Step-by-step explanation:
Partition of a Segment
The length of segment directed from A(xa,ya) to B(xb,yb) can be decomposed in its x,y coordinates:
[tex]x_{AB}=x_b-x_a[/tex]
[tex]y_{AB}=y_b-y_a[/tex]
If a point E is to be in the segment and partition it into a ratio 1:2, then
\displaystyle \frac{x_{AE}}{{x_{EB}}=\frac{y_{AE}}{{y_{EB}}=\frac{1}{2}
But
[tex]x_{AE}=x_e-x_a[/tex]
[tex]y_{AE}=y_e-y_a[/tex]
[tex]x_{EB}=x_b-x_e[/tex]
[tex]y_{EB}=y_b-y_e[/tex]
Then we set the equation:
[tex]\frac{x_{AE}}{{x_{EB}}=\frac{1}{2}[/tex]
[tex]2(x_e-x_a)=x_b-x_e[/tex]
Operating and rearranging
[tex]3x_e=x_b+2x_a[/tex]
Solving
[tex]\displaystyle x_e=\frac{x_b+2x_a}{3}=\frac{12+6}{3}[/tex]
[tex]x_e=6[/tex]
Similarily
[tex]\displaystyle y_e=\frac{y_b+2y_a}{3}=\frac{-5+2}{3}[/tex]
[tex]y_e=-1[/tex]
The coordinates of E are (6,-1)
