Compute distance, midpoint, and slope from coordinates, and use slope to test whether lines are parallel or perpendicular.
Coordinate geometry drops familiar shapes onto the x-y plane, where formulas take over for rulers and protractors. The distance formula finds the length of any segment from its endpoints' coordinates: distance = sqrt((x2-x1)^2 + (y2-y1)^2) -- really the Pythagorean theorem in disguise, treating the horizontal and vertical gaps between the two points as the legs of a right triangle and the segment itself as the hypotenuse. The midpoint formula finds the point exactly halfway between two endpoints by averaging their coordinates separately: midpoint = ((x1+x2)/2, (y1+y2)/2).
Slope measures steepness: slope = (y2-y1)/(x2-x1), the change in y over the change in x between any two points on a line. Slope also reveals how two lines relate without graphing them: parallel lines have the exact same slope, and perpendicular lines have slopes that are negative reciprocals of each other (flip the fraction, then switch the sign) -- multiply a perpendicular pair of slopes together and you always get -1. A horizontal line has slope 0 and a vertical line has an undefined slope; a horizontal and a vertical line are always perpendicular to each other, even though the "multiply to -1" shortcut does not literally apply to zero and undefined.
Mastery looks like: They compute distance, midpoint, and slope from coordinates and use slope to determine whether two lines are parallel, perpendicular, or neither.
Common stumbles: Subtracting coordinates in an inconsistent order between the x-values and y-values; flipping the slope for a perpendicular line but forgetting to also switch its sign.
See it visually
Interactive, spoken-aloud tools that show this idea in motion.