Overview

Coordinate geometry converts geometry to algebra. The distance and midpoint formulas are the core tools, and slope links geometry to lines.

Key Ideas

• Distance: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
• Midpoint: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.
• Slope: $m = \frac{y_2-y_1}{x_2-x_1}$.
• Horizontal lines have slope $0$; vertical lines have undefined slope.

Core Skills

Translate Geometry to Coordinates

Set up points, then use distance and midpoint formulas directly. Keep track of signs when subtracting coordinates.

Use Slope for Parallel/Perpendicular

Parallel lines share slope. Perpendicular lines have slopes that are negative reciprocals (when defined).

Avoid Square Roots Early

For comparisons, use squared distances to avoid extra radicals.

Worked Example

Find the midpoint of $(0,0)$ and $(6,4)$.

The midpoint is $(3,2)$.

More Examples

Example 1: Distance

Find the distance between $(1,2)$ and $(7,5)$.

$d = \sqrt{(6)^2 + (3)^2} = \sqrt{45} = 3\sqrt{5}$.

Example 2: Slope

Find the slope of the line through $(2,-1)$ and $(6,7)$.

$m = (7-(-1))/(6-2) = 8/4 = 2$.

Example 3: Compare Lengths

Which is longer, $AB$ with $A(0,0),B(3,4)$ or $CD$ with $C(1,1),D(5,6)$?

$AB^2 = 3^2+4^2=25$ and $CD^2=4^2+5^2=41$, so $CD$ is longer.

Strategy Checklist

• Label coordinates clearly before plugging into formulas.
• Use squared distances for comparisons.
• Check for vertical lines before using slope.
• Keep fractions exact when possible.

Common Pitfalls

• Forgetting to square both coordinate differences in distance.
• Dividing each coordinate by 2 before adding.
• Using slope for vertical lines (undefined).

Practice Problems