Question

What are the coordinates of point B on \overline{AC} such that AB = 1/2 BC

Answer 1

C.
`B(x,y) = \left(3,-(16)/(3) \right)`

Explanation:

Let
`A(x,y) = (6,-7)`,
`B(x,y) = (x,y)` and
`C(x,y) = (-3,-2)`. From statement we know that
`AB = (1)/(2)\cdot BC`, which is equivalent to the following linear algebraic formula:

`B(x,y) -A(x,y) = (1)/(2)\cdot [C(x,y)-B(x,y)]` (1)

`B(x,y)-A(x,y) = (1)/(2)\cdot C(x,y)-(1)/(2)\cdot B(x,y)`

`(3)/(2)\cdot B(x,y) = (1)/(2)\cdot C(x,y)+A(x,y)`

`B(x,y) = (1)/(3)\cdot C(x,y) +(2)/(3)\cdot A(x,y)` (2)

Then, the coordinates of point B on AC are:

`B(x,y) = (1)/(3)\cdot (-3,-2)+(2)/(3)\cdot (6,-7)`

`B(x,y) = \left(-1, -(2)/(3)\right)+\left(4, -(14)/(3) \right)`

`B(x,y) = \left(3,-(16)/(3) \right)`

Which means that correct answer is C.