Coordinate geometry connects algebra and geometry by placing points on a grid. The distance between two points (x₁, y₁) and (x₂, y₂) is given by d = √[(x₂ − x₁)² + (y₂ − y₁)²], a formula derived directly from the Pythagorean Theorem. The midpoint of a segment with these endpoints is M = ((x₁ + x₂)/2, (y₁ + y₂)/2), which gives the point exactly halfway between them. The slope of a line through two points is m = (y₂ − y₁)/(x₂ − x₁), and slope measures both the steepness and the direction of a line. Lines can be written in several useful forms. The slope-intercept form y = mx + b makes the slope m and y-intercept b immediately visible. The point-slope form y − y₁ = m(x − x₁) is convenient when you know a slope and one point. The standard form Ax + By = C, where A, B, and C are integers with A non-negative, makes it easy to read off intercepts. To find the equation of a line through two points, first calculate the slope using the slope formula, then substitute m and one of the points into point-slope form and simplify.
Several other formulas rely on coordinates. The area of a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃) is A = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|, known as the Shoelace formula for three points. A circle with center (h, k) and radius r has the standard-form equation (x − h)² + (y − k)² = r². The shortest distance from a point (x₀, y₀) to a line Ax + By + C = 0 is d = |Ax₀ + By₀ + C| / √(A² + B²), which always measures the perpendicular distance.
Transformations move figures around the coordinate plane. A translation slides every point the same distance in the same direction, mapping (x, y) to (x + a, y + b) and preserving size, shape, and orientation. A reflection flips a figure over a line of reflection, creating a mirror image that preserves size and shape but reverses orientation. Reflecting (x, y) over the x-axis gives (x, −y); reflecting over the y-axis gives (−x, y). A rotation turns a figure around a fixed center by a specified angle, preserving size and shape. Rotating (x, y) by 90° counterclockwise about the origin gives (−y, x), while a 90° clockwise rotation gives (y, −x). A dilation enlarges or reduces a figure by a scale factor k from a center point, mapping (x, y) to (kx, ky) when the center is the origin.
When two lines are in the same plane and never intersect, no matter how far they are extended, they are parallel and share the same slope. Lines that meet at a 90° angle are perpendicular, and the product of their slopes is −1; if one slope is m, the perpendicular slope is −1/m. A transversal cuts across two lines and creates special angle pairs: corresponding angles (same side of transversal, same position relative to the lines) are congruent when the lines are parallel; alternate interior angles (opposite sides of transversal, between the lines) are congruent; alternate exterior angles (opposite sides of transversal, outside the lines) are congruent; and co-interior or same-side interior angles (same side of transversal, between the lines) are supplementary, summing to 180°. Any one of these angle relationships can be used in reverse to prove that two lines are parallel.
Trigonometry extends right-triangle work with ratios of side lengths. For an angle θ in a right triangle, the sine is sin θ = opposite / hypotenuse, the cosine is cos θ = adjacent / hypotenuse, and the tangent is tan θ = opposite / adjacent, which also equals sin θ / cos θ. The mnemonic SOH-CAH-TOA captures these: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. The reciprocal functions extend these ideas: cosecant csc θ = 1/sin θ, secant sec θ = 1/cos θ, and cotangent cot θ = 1/tan θ. A fundamental identity, sin²θ + cos²θ = 1, follows from the Pythagorean Theorem applied to the unit circle. To find an unknown side, identify the angle and which sides are involved, choose the matching ratio, set up the equation, and solve. For instance, knowing angle θ and the hypotenuse gives opposite = hypotenuse × sin θ, while knowing an angle and the opposite side gives hypotenuse = opposite / sin θ. Together, coordinate geometry, transformations, and trigonometry provide a powerful toolkit for analyzing geometric figures from multiple perspectives.