Coordinate tricks that make the plane click

Slope = rise/run = (y₂-y₁)/(x₂-x₁). Parallel: equal slopes. Perpendicular: negative reciprocals.

Slope Formula — Positive slope: up left to right. Negative: down. Zero: horizontal. Undefined: vertical. Parallel lines: equal slopes. Perpendicular lines: slopes that multiply to -1 (e.g., 2 and -½).

Distance = √[(x₂-x₁)² + (y₂-y₁)²] — Pythagoras in the coordinate plane

Distance Formula — Square the x-difference, add the squared y-difference, take the square root. It IS the Pythagorean theorem — the horizontal and vertical gaps are the two legs.

Midpoint = average of coordinates: ((x₁+x₂)/2, (y₁+y₂)/2)

Midpoint Formula — The midpoint is literally the average position. x-mid = (x₁+x₂)/2. y-mid = (y₁+y₂)/2. Used to find centers, bisectors, and medians of triangles.

y = mx + b: m = slope, b = y-intercept

Slope-Intercept Form — m is slope (steepness and direction). b is y-intercept (where line crosses y-axis). To graph: start at (0, b), then go rise/run. Parallel lines share m. Perpendicular: m₁ × m₂ = -1.

Equation of a circle: (x-h)² + (y-k)² = r² where (h,k) is center and r is radius

Equation of a Circle — Center (h, k), radius r. Expand to get general form: x² + y² + Dx + Ey + F = 0. To convert back to standard form, complete the square for x and y separately.

Standard form of a line: Ax + By = C. Convert to slope-intercept: solve for y.

Standard Form of a Line — Standard form: Ax + By = C where A, B, C are integers and A ≥ 0. Easy to find x and y intercepts: set y=0 for x-intercept, set x=0 for y-intercept. Converting to slope-intercept: By = -Ax + C → y = (-A/B)x + C/B. Point-slope form: y - y₁ = m(x - x₁) — useful when you have a point and slope.

Quadrants: I (positive x, positive y) upper right. II (negative x, positive y) upper left. III (negative x, negative y) lower left. IV (+,-) lower right.

Coordinate Plane Quadrants — Quadrant I: x>0, y>0 (upper right). Quadrant II: x<0, y>0 (upper left). Quadrant III: x<0, y<0 (lower left). Quadrant IV: x>0, y<0 (lower right). Memory: start upper right, go counterclockwise. Points on axes are not in any quadrant.

Perpendicular bisector of a segment: passes through midpoint at 90°. All points equidistant from both endpoints.

Perpendicular Bisector — The perpendicular bisector of segment AB: passes through the midpoint of AB at a 90° angle. Every point on the perpendicular bisector is equidistant from A and B. To find it: (1) find midpoint, (2) find slope of AB, (3) take negative reciprocal for perpendicular slope, (4) write equation through midpoint.

Classify triangles by vertices: find side lengths with distance formula, find slopes to check right angles

Coordinate Geometry Triangle Classification — Scalene: all three sides different lengths. Isosceles: two sides equal (use distance formula). Equilateral: all three sides equal. Right: check if two sides are perpendicular (slopes are negative reciprocals). Right isosceles: both conditions. Use distance formula for side lengths, slope for angle types.

Locus: set of all points satisfying a condition. Circle is locus of points equidistant from center.

Locus Problems — Locus: all points satisfying a given condition. Locus equidistant from two points: perpendicular bisector. Locus equidistant from two parallel lines: parallel line between them. Locus equidistant from a point (circle center): circle. Locus equidistant from two intersecting lines: angle bisectors.

Slope of horizontal line = 0. Vertical line = undefined. Parallel lines have same slope.

Slope Special Cases — Horizontal line (y = k): slope = 0, zero rise. Vertical line (x = k): slope = undefined, zero run (can't divide by zero). Two lines are parallel if slopes are equal (and they're different lines). Two lines are perpendicular if slopes are negative reciprocals: m₁ × m₂ = -1. Horizontal ⊥ vertical: 0 × undefined = special case.

Transformations in coordinate geometry: translate (add), reflect (negate), rotate (use rules), dilate (multiply)

Coordinate Transformations Summary — Translation by (a,b): (x,y) → (x+a, y+b). Reflection over x-axis: (x,y) → (x,-y). Reflection over y-axis: (x,y) → (-x,y). Reflection over y=x: (x,y) → (y,x). Rotation 90° CCW: (x,y) → (-y,x). Dilation by k from origin: (x,y) → (kx,ky).