perpendiculum

Perpendiculum is a historical term in geometry, Latin for "a perpendicular." In classic Euclidean geometry it refers to the perpendicular line segment drawn from a given point to a given line, or to the length of that segment, i.e., the perpendicular distance from the point to the line. The perpendiculum is unique: through any point not on the line there exists exactly one line perpendicular to the reference line, and the portion of that line between the point and the line is the perpendiculum. The point where it meets the line is called the foot of the perpendicular.

In analytic form, if the reference line is ax + by + c = 0 and the point is (x0, y0), the length of the perpendiculum equals |ax0 + by0 + c| / sqrt(a^2 + b^2). The equation of the perpendicular line through (x0, y0) has a direction perpendicular to the original line, with slope -a/b (when b ≠ 0). The perpendiculum is used to compute the shortest distance from a point to a line and to relate to triangle heights and area calculations, since the area equals base times height, i.e., base length times perpendiculum length divided by 2.

Today, the term perpendiculum is largely historical or found in older Latin texts; contemporary mathematical language typically uses "perpendicular segment" or "perpendicular distance" to describe the same concept. See also: perpendicular, perpendicular distance, foot of the perpendicular.