Coordinate Geometry and Transformations
Coordinate geometry asks you to prove geometric facts algebraically. Memorize the tools: midpoint formula, distance formula, slope formula, and the slope criteria for parallel (equal slopes) and perpendicular (negative reciprocal slopes, product -1). Vertical and horizontal lines are special cases: vertical lines have undefined slope, and a vertical line is perpendicular to a horizontal line even though the negative-reciprocal rule does not apply cleanly.
For transformations, know the coordinate rules. Translations add to coordinates. Reflections: over the x-axis, (x, y) becomes (x, -y); over the y-axis, (-x, y); over y = x, (y, x); over y = -x, (-y, -x). Rotations about the origin: 90 degrees counterclockwise maps (x, y) to (-y, x); 180 degrees maps to (-x, -y); 270 degrees counterclockwise (or 90 clockwise) maps to (y, -x). Dilations multiply both coordinates by the scale factor about the center. Rigid motions (translations, reflections, rotations) preserve distance and angle measure and produce congruent figures; dilations produce similar figures.
Traps: rotating the wrong direction, reflecting over the wrong axis, and forgetting that a composition of transformations is performed right to left. Also watch for questions asking whether a transformation maps a figure onto itself (symmetry).
Worked example. Quadrilateral vertices A(1, 2), B(4, 2), C(4, 6), D(1, 6). Slopes: AB is horizontal (0), BC is vertical (undefined), so AB is perpendicular to BC; likewise all four corners are right angles, and side lengths are 3, 4, 3, 4, so the figure is a rectangle.