Skill: Determine an endpoint from a midpoint and another endpoint

The midpoint of a segment is found by averaging the corresponding coordinates of its endpoints.

If the midpoint is $(M(x_m,y_m))$ and one endpoint is $(A(x_1,y_1))$, then the other endpoint $B$ is

$$B(2x_m-x_1,\;2y_m-y_1).$$

This works because the midpoint must be halfway between the two endpoints in both the horizontal and vertical directions.

Example: Find endpoint $B$ if the midpoint is $(M(4,3))$ and one endpoint is $(A(1,7))$.

1. Use the formula:

$$B(2x_m-x_1,\;2y_m-y_1)$$

2. Substitute the coordinates:

$$B(2(4)-1,\;2(3)-7)$$

3. Simplify each coordinate:

$$B(8-1,\;6-7)$$ $$B(7,-1)$$

Therefore, the missing endpoint is

$$\boxed{B(7,-1)}.$$

You can check the answer by averaging the endpoints:

$$\left(\frac{1+7}{2},\frac{7+(-1)}{2}\right) = \left(\frac{8}{2},\frac{6}{2}\right) = (4,3),$$

which is the given midpoint.