SAT Coordinate Geometry: Lines & Parabolas

Coordinate geometry questions appear 3–5 times per SAT, covering lines, midpoints, distances, and parabolas on the xy-plane. You'll work with slope (rise/run), slope-intercept form (y = mx + b), point-slope form, parallel and perpendicular lines, midpoint formula, and distance formula.

The SAT also connects coordinate geometry to systems of equations — finding where two lines intersect is a system of equations problem.

Slope from two points: (y₂−y₁)/(x₂−x₁). Label the two points before substituting to avoid sign errors — this is where most slope mistakes originate.

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals: if one line has slope 2/3, the perpendicular line has slope −3/2. Both negate and flip.

Midpoint: average both coordinates: ((x₁+x₂)/2, (y₁+y₂)/2). For distance, use the Pythagorean theorem: √((x₂−x₁)² + (y₂−y₁)²). If the points share an x or y coordinate, just subtract.

Swapping rise and run in slope calculations. Rise is the change in y; run is the change in x. Writing the formula helps: always put y-values in the numerator.

Getting the perpendicular slope wrong by only negating or only flipping — you must do both. Two lines with slopes 3 and −1/3 are perpendicular; slopes 3 and −3 are not.

Using the distance formula when the points share a coordinate. If both points have y = 4, the distance is just |x₂ − x₁|.

Practice identifying which formula you need in the first five seconds: slope, distance, or midpoint each have a distinct problem structure. Faster recognition means more time for solving.

ScoreBooster's Geometry sessions mix coordinate geometry with angle and area questions so you practice switching between algebraic and visual reasoning — exactly how the SAT combines these.