Answer :
Answer:
The position of point B is:
[tex](x_{3}, y_{3}) = \left(\frac{2}{3}\cdot x_{2} + \frac{1}{3} \cdot x_{1}, \frac{2}{3}\cdot y_{2} + \frac{1}{3} \cdot y_{1} \right)[/tex]
Step-by-step explanation:
Let be [tex]A = (x_{1}, y_{1})[/tex], [tex]B = (x_{3}, y_{3})[/tex] and [tex]C = (x_{2}, y_{2})[/tex]. The ratio is:
[tex]\frac{(x_{3}-x_{1},y_{3}-y_{1})}{(x_{2}-x_{3},y_{2}-y_{3})} = 2[/tex]
After some algebraic handling:
[tex](x_{3}-x_{1}, y_{3}-y_{1}) = 2 \cdot (x_{2}-x_{3},y_{2}-y_{3})[/tex]
[tex](x_{3}-x_{1}, y_{3}-y_{1}) = (2\cdot x_{2} - 2\cdot x_{3}, 2\cdot y_{2} - 2\cdot y_{3})[/tex]
[tex](3\cdot x_{3}, 3\cdot y_{3}) = (2\cdot x_{2} + x_{1}, 2\cdot y_{2} + y_{1})[/tex]
[tex]3\cdot (x_{3}, y_{3}) = (2\cdot x_{2} + x_{1}, 2\cdot y_{2} + y_{1})[/tex]
[tex](x_{3},y_{3}) = \frac{1}{3}\cdot (2\cdot x_{2}+x_{1}, 2\cdot y_{2}+y_{1})[/tex]
The position of point B is:
[tex](x_{3}, y_{3}) = \left(\frac{2}{3}\cdot x_{2} + \frac{1}{3} \cdot x_{1}, \frac{2}{3}\cdot y_{2} + \frac{1}{3} \cdot y_{1} \right)[/tex]