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Analytic Geometry: (lesson 1 of 4)

The distance between two points in the plane is the length of the line segment joining the two points. If the points have Cartesian coordinates $(x_1, x_2)$ and $(y_1, y_2)$, then

$$ d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} $$

If the points have polar coordinates $(r_1, \theta_1)$ and $(r_2, \theta_2)$, then

$$ d = \sqrt{{r_1}^2 + {r_2}^2 + 2 r_1 r_2 \cos (\theta_1 - \theta_2)} $$

The midpoint of two points $(x_1, x_2)$ and $(y_1, y_2)$ is

$$ \left( \frac{x_1 + x_2}{2}, \frac{y_1 - y_2}{2} \right) $$