retas

Retas is the Portuguese term for straight lines in geometry. A reta is the set of all points extending infinitely in both directions along a single direction and with zero curvature. It is uniquely determined by two distinct points, by a point and a direction, or by a linear equation. In the plane, lines can be written as y = mx + b, where m is the slope, or in standard form ax + by + c = 0. The slope m describes how steep the line is; vertical lines have undefined slope.

Two retas are parallel if they have the same slope (or are coincident); they are perpendicular if the product of their slopes is -1, provided neither is vertical. In three-dimensional space, lines may be parallel, intersect, or be skew (neither intersecting nor parallel).

The intersection of two non-parallel retas is a single point, obtainable by solving their equations. The distance from a point (x0, y0) to a reta ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a^2 + b^2). For parallel retas ax + by + c1 = 0 and ax + by + c2 = 0, the distance between them is |c2 − c1| / sqrt(a^2 + b^2).

In Portuguese geometry, retas are distinguished from segmentos de reta (line segments) and raios (rays). Retas form a foundational concept used across mathematics, physics, engineering, computer graphics, and design.