Perpendicular

In geometry, perpendicularity describes the relationship between two geometric objects that meet at a right angle—an angle of 90∘90^\circ90∘ or π2\tfrac{\pi}{2}2π​ radians. The concept applies broadly to intersections between lines, line segments, rays, planes and certain conic elements. The symbol ⊥\perp⊥ is used to represent this relationship, and the term perpendicular may also serve as a noun, referring to a line dropped at a right angle to another line or plane.

Basic Definition and Orthogonality

Two lines are considered perpendicular if they intersect and form congruent adjacent angles that together make a straight angle. The condition is symmetric: if line aaa is perpendicular to line bbb, then line bbb is perpendicular to line aaa. More generally, perpendicularity is a specific example of orthogonality, a broader mathematical principle that includes relationships between vectors, curves and surfaces. In advanced geometry, the term may therefore describe orthogonality between entities such as a surface and its normal.
Examples in everyday spatial reasoning include the north–south and east–west axes on a compass, which intersect at right angles, dividing the circle into four equal quadrants.

Perpendicular Line Segments, Rays and Planes

The concept extends readily to finite geometric objects. A line segment AB‾\overline{AB}AB is perpendicular to another segment CD‾\overline{CD}CD if their extended lines meet perpendicularly. A line is perpendicular to a plane if it intersects the plane and is perpendicular to every line in the plane passing through the point of intersection. Two planes are perpendicular if their dihedral angle is 90∘90^\circ90∘.
A standard construction involves the foot of a perpendicular: given a point PPP and a line lll, the unique line through PPP perpendicular to lll meets the line at a point FFF. This point is called the foot of the perpendicular from PPP to lll.

Constructing a Perpendicular

Classical compass-and-straightedge constructions offer methods for drawing perpendiculars:

• From a point on a line: create two points equidistant on the line from the chosen point, then draw arcs from these two points to locate a pair of intersection points, and finally connect the new point to the original point to form a perpendicular.
• From a point off a line: apply the same intersecting arc method or use Thales’ theorem, which states that an angle inscribed in a semicircle is a right angle.

Practical methods such as the 3–4–5 triangle rule, based on the Pythagorean theorem, help establish right angles in fieldwork.

Perpendicularity and Parallel Lines

If two lines aaa and bbb are both perpendicular to a third line ccc, then aaa and bbb are parallel, a consequence of the Euclidean parallel postulate. The relationship also implies congruence of corresponding angles: vertical angles and alternate interior angles remain equal when a transversal crosses parallel lines. Conversely, if one line is perpendicular to another, it is also perpendicular to every line parallel to the second.

Analytic Geometry and Perpendicular Slopes

In the coordinate plane, two lines with slopes m1m_1m1​ and m2m_2m2​ are perpendicular if their slopes satisfy
m1m2=−1.m_1 m_2 = -1.m1​m2​=−1.
This condition arises from the dot product of their direction vectors being zero—a hallmark of orthogonality. Vertical and horizontal lines represent limiting cases in which one slope is zero and the other undefined or infinite.

Circles and Other Conic Sections

Perpendicularity appears in several classical results concerning conics:

• Circles: a diameter is perpendicular to the tangent at its endpoint. A line through a circle’s centre that bisects a chord is perpendicular to the chord. Thales’ theorem states that any triangle inscribed in a semicircle has a right angle opposite the diameter.
• Ellipses: the major and minor axes are perpendicular to each other and to the tangent where they meet the ellipse.
• Parabolas: the axis of symmetry is perpendicular to the directrix and to the tangent at the vertex. The orthoptic property states that two perpendicular tangents intersect on the directrix.
• Hyperbolas: the transverse and conjugate axes of a hyperbola relate orthogonally, and in a rectangular hyperbola the asymptotes are perpendicular.