Question

If B is the midpoint of AC and A is (4,1) and B is (-3,5), what are the coordinates of C?

Answer 1

Solving for coordinates of the endpoint given the midpoint and the other endpoint

Answer:

`(-10,9)`

Explanation:

Recall that if you have the two points,
`(x_1,y_1)` and
`(x_2,y_2)`, the midpoint between them is
`((x_1 +x_2)/(2), (y_1 +y_2)/(2))\\`.

We let
`x_c` be the
`x`-coordinate of point
`C` and
`y_c` for the
`y`-coordinate of point
`C`.

The problem tells us that point
`B` is the midpoint of
`\overline{AC}`. This means that the point
`(-3,5)` is the same point as
`((4 +x_c)/(2), (1 +y_c)/(2))\\`.

Solving for
`x_c`:

`(4 +x_c)/(2) = -3 \\ (4 +x_c)/(2) \cdot 2 = -3 \cdot 2 \\ 4 +x_c = -6 \\ 4 +x_c -4 = -6 -4 \\ x_c = -10`

Solving for
`y_c`:

`(1 +y_c)/(2) = 5 \\ (1 +y_c)/(2) \cdot 2 = 5 \cdot 2 \\ 1 +y_c = 10 \\ 1 +y_c -1 = 10 -1 \\ y_c = 9`

The coordinates of point
`C` is
`(-10,9)`.