### Theory:

Problem $$1$$:

Find the mid-point of the vertices $$A(1$$, $$1)$$ and $$B(-1$$, $$-1)$$.

Given:

The end points of the line segment are $$A(1$$, $$1)$$ and $$B(-1$$, $$-1)$$.

$$x_1 = 1$$

$$x_2 = -1$$

$$y_1 = 1$$

$$y_2 = -1$$

Let $$M$$ be the mid-point of the line segment.

$$\text{Midpoint} = (\frac{x_1+x_2}{2}$$, $$\frac{y_1+y_2}{2})$$

$$= (\frac{1 - 1}{2}$$, $$\frac{1 - 1}{2})$$

$$= (\frac{0}{2}$$, $$\frac{0}{2})$$

$$M = (0$$, $$0)$$

Problem $$2$$:

Find the mid-point of the vertices $$A(9$$, $$2)$$ and $$B(4$$, $$5)$$.

Given:

The end points of the line segment are $$A(9$$, $$2)$$ and $$B(4$$, $$3)$$.

$$x_1 = 9$$

$$x_2 = 4$$

$$y_1 = 2$$

$$y_2 = 3$$

Let $$M$$ be the mid-point of the line segment.

$$\text{Midpoint} = (\frac{x_1 + x_2}{2}$$, $$\frac{y_1 + y_2}{2})$$

$$= (\frac{9 + 4}{2}$$, $$\frac{2 + 3}{2})$$

$$M = (\frac{13}{2}$$, $$\frac{5}{2})$$