Overview

Coordinate geometry translates geometry into algebra. Use slopes for collinearity and the distance formula for lengths.

Key Ideas

• Distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
• Midpoint formula: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.
• Parallel lines have equal slopes, perpendicular lines have slopes that multiply to $-1$.

Core Skills

Check Collinearity

Compare slopes of pairs of points or use the area of a triangle to be zero.

Use Vector Differences

Vectors like $\overrightarrow{AB}$ simplify distance and parallelism checks.

Handle Vertical Lines

Vertical lines have undefined slope; use $x=c$ and avoid slope formulas.

Worked Example

Find the equation of the line through $(2,3)$ and $(6,11)$.

Slope is $\frac{11-3}{6-2}=2$, so the line is $y-3=2(x-2)$ or $y=2x-1$.

More Examples

Example 1: Collinearity

Are $(0,0)$, $(2,3)$, and $(4,6)$ collinear?

Slope from $(0,0)$ to $(2,3)$ is $3/2$, and to $(4,6)$ is $6/4=3/2$, so yes.

Example 2: Perpendicular Slope

Find the slope of a line perpendicular to $y=3x+4$.

The slope is $-1/3$.

Example 3: Midpoint in Coordinate Proof

Find the midpoint of $(1,5)$ and $(7,1)$.

The midpoint is $(4,3)$.

Strategy Checklist

• Compute slopes carefully and simplify.
• Use vectors to avoid repeated subtraction.
• Treat vertical lines separately.

Practice Problems