Geometry / Unit 1: Foundations of Geometry
1.3 Midpoint and Distance Formulas
Notes
The midpoint of a segment splits it into two congruent halves. On the coordinate plane, average the coordinates.
- Midpoint formula: $\displaystyle M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
- Distance formula: $\displaystyle d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. It comes from the Pythagorean theorem.
- Example: $A(1, 2)$ and $B(7, 10)$. Midpoint $(4, 6)$. Distance $\displaystyle \sqrt{6^2 + 8^2} = \sqrt{100} = 10$.
- To find a missing endpoint, double the midpoint and subtract the known endpoint.
Practice
What is the midpoint of the segment from $(-2, 5)$ to $(6, -1)$?
$\displaystyle \left(\frac{-2 + 6}{2}, \frac{5 + (-1)}{2}\right) = (2, 2)$.
What is the distance between $(1, -3)$ and $(4, 1)$?
$\displaystyle \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
$M(3, 4)$ is the midpoint of $\overline{AB}$, and $A$ is $(1, -2)$. Find $B$.
$B = (2 \cdot 3 - 1, 2 \cdot 4 - (-2)) = (5, 10)$.
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