Answer:
C. [tex]B(x,y) = \left(3,-\frac{16}{3} \right)[/tex]
Step-by-step explanation:
Let [tex]A(x,y) = (6,-7)[/tex], [tex]B(x,y) = (x,y)[/tex] and [tex]C(x,y) = (-3,-2)[/tex]. From statement we know that [tex]AB = \frac{1}{2}\cdot BC[/tex], which is equivalent to the following linear algebraic formula:
[tex]B(x,y) -A(x,y) = \frac{1}{2}\cdot [C(x,y)-B(x,y)][/tex] (1)
[tex]B(x,y)-A(x,y) = \frac{1}{2}\cdot C(x,y)-\frac{1}{2}\cdot B(x,y)[/tex]
[tex]\frac{3}{2}\cdot B(x,y) = \frac{1}{2}\cdot C(x,y)+A(x,y)[/tex]
[tex]B(x,y) = \frac{1}{3}\cdot C(x,y) +\frac{2}{3}\cdot A(x,y)[/tex] (2)
Then, the coordinates of point B on AC are:
[tex]B(x,y) = \frac{1}{3}\cdot (-3,-2)+\frac{2}{3}\cdot (6,-7)[/tex]
[tex]B(x,y) = \left(-1, -\frac{2}{3}\right)+\left(4, -\frac{14}{3} \right)[/tex]
[tex]B(x,y) = \left(3,-\frac{16}{3} \right)[/tex]
Which means that correct answer is C.
