# Geometry of the Plane

## Worked Examples

## Other Formulas

**Distance.**The formula for calculating the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $$ \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}. $$

**Proof.**We can prove this formula using the Pythagorean Theorem.

One can observe that the distance solves the equation:
$$
{\rm distance}^2 = |x_1-x_2|^2 + |y_1-y_2|^2
$$
so it takes the formula above.

**Slope of a line segment.**The formula for calculating the slope of a line segment between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $$ m=\frac{y_1-y_2}{x_1-x_2}. $$

**Midpoint.**The formula for calculating the midpoint between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $$ \left( \frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) $$