Perpendicular

The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees.

For other uses, see Perpendicular (disambiguation).

In geometry, two lines or planes (or a line and a plane) are considered perpendicular (or orthogonal) to each other if they form congruent adjacent angles (a T-shape). The term may be used as a noun or adjective. Thus, as illustrated, the line AB is the perpendicular to CD through the point B.

By definition, a line is infinitely long, and strictly speaking AB and CD in this example represent line segments of two infinitely long lines. Hence the line segment AB does not have to intersect line segment CD to be considered perpendicular lines, because if the line segments are extended out to infinity, they would still form congruent adjacent angles.

If a line is perpendicular to another as shown, all of the angles created by their intersection are called right angles (right angles measure π/2 radians, or 90°). Conversely, any lines that meet to form right angles are perpendicular.

In a coordinate plane, perpendicular lines have opposite reciprocal slopes, which means that the product of their slopes is -1. A horizontal line has slope equal to zero while the slope of a vertical line is described as undefined or sometimes ±infinity. Two lines that are perpendicular would be denoted as AB$\perp$CD.

Construction of the perpendicular

Construction of the perpendicular (blue) to the line AB through the point P.

To make the perpendicular to the line AB through the point P using compass and straightedge, proceed as follows (see figure):

• Step 1 (red): construct a circle with center at P to create points A' and B' on the line AB, which are equidistant from P.
• Step 2 (green): construct circles centered at A' and B' having equal radius. Let Q and R be the points of intersection of these two circles.
• Step 3 (blue): connect Q and R to construct the desired perpendicular PQ.

To prove that the PQ is perpendicular to AB, use the SSS congruence theorem for ' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the SAS congruence theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal.

In relationship to parallel lines

Lines a and b are parallel, as shown by the tick marks, and are cut by the transversal line c.

If two lines (a and b) are both perpendicular to a third line (c), all of the angles formed along the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.

In the figure at the right, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because vertical angles are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines a and b are parallel, any of the following conclusions leads to all of the others:

• One of the angles in the diagram is a right angle.
• One of the orange-shaded angles is congruent to one of the green-shaded angles.
• Line 'c' is perpendicular to line 'a'.
• Line 'c' is perpendicular to line 'b'.

Perpendicular symbol

The perpendicular symbol is $\perp$ . For example, $AB \perp CD$ indicates that line AB is perpendicular to line CD.

In the Unicode character set, the perpendicular sign has the codepoint U+27C2 and is part of the Miscellaneous Mathematical Symbols-A range. It looks similar to the up tack symbol (U+22A5).

Graph of functions

In 2-dimension plane, right angles can be formed by two intersected lines which the product of their slopes equals to −1. More precisely, defining two linear functions: y₁ = a₁x + b₁ and y₂ = a₂x + b₂, the graph of the functions will be perpendicular and will make four right angles where the lines intersect if and only if a₁a₂ = −1. However, this method cannot be used if the slope is zero or undefined (the line is parallel to an axis).

For another method, let the two linear functions: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. The lines will be perpendicular if and only if a₁a₂ + b₁b₂ = 0. This method is simplified from the dot product (or generally, inner product) of vectors. In particular, two vectors are considered orthogonal if their inner product is zero.

See also

• Orthogonality
• Perpendicular component (of a vector)
• Surface normal
• Parallel (geometry)

External links

• Definition: perpendicular With interactive animation
• How to draw a perpendicular bisector of a line with compass and straight edge Animated demonstration
• How to draw a perpendicular at the endpoint of a ray with compass and straight edge Animated demonstration