What is Perpendicular?

Information about Perpendicular

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Fig. 1: The line AB is perpendicular to the line CD, because the two angles it creates (indicated in orange and blue, respectively) are each 90 degrees.


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. The term may be used as a noun or adjective. Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B.

If a line is bending to another as in Figure 1, all of the angles created by their intersection are called right angles (right angles measure ½π radians, or 90°). Conversely, any lines that meet to form right angles are perpendicular. The line AB does not have to end at B to be considered perpendicular.

In a co-ordinate plane, perpendicular lines have opposite reciprocal slopes. Horizontal and vertical lines have zero and positive/negative infinity.

Numerical criteria

In terms of slopes

In a Cartesian coordinate system, two straight lines and may be described by equations.
as long as neither is vertical. Then and are the slopes of the two lines. The lines and are perpendicular if and only if the product of their slopes is -1, or if .

The perpendiculars to vertical lines are always horizontal lines, and the perpendiculars to horizontal lines are always vertical lines. All horizontal lines are perpendicular to all vertical lines; that is, for any horizontal line and horizontal line , where and are constants, .

Construction of the perpendicular

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Fig. 2: Construction of the perpendicular (blue) to the line AB through the point P.
To construct the perpendicular to the line AB through the point P using compass and straightedge, proceed as follows (see Figure 2).
  • 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', both passing through P. Let Q be the other point of intersection of these two circles.
  • Step 3 (blue): connect P and Q to construct the desired perpendicular PQ.
To prove that the PQ is perpendicular to AB, use the SSS congruence theorem for triangles QPA' 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

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Fig. 3: Lines a and b are parallel, as shown by the tick marks, and are cut by the transversal line c.
As shown in Figure 3, if two lines (a and b) are both perpendicular to a third line (c), all of the angles formed on 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 Figure 3, 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'.

See also

External links

Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
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line can be described as an ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points.
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plane is a two-dimensional manifold or surface that is perfectly flat. Informally it can be thought of as an infinitely vast and infinitesimally thin sheet oriented in some space.
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congruent if one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. In less formal language, two sets are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated,
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Trigonometry (from Greek trigōnon "triangle" + metron "measure"[1]), informally called trig, is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled
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The Angles is a modern English word for a Germanic-speaking people who took their name from the cultural ancestor of Angeln, a modern district located in Schleswig-Holstein, Germany.
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Examples
A proper or common noun can co-occur with an article or an attributive adjective. Verbs and adjectives can't. As usual, a `*' in front of an example means that this example is ungrammatical.
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    In grammar, an adjective is a word whose main syntactic role is to modify a noun or pronoun (called the adjective's subject), giving more information about what the noun or pronoun refers to.
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    right angle is an angle of 90 degrees, corresponding to a quarter turn (that is, a quarter of a full circle). It can be defined as the angle such that twice that angle amounts to a half turn, or 180° [1].
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    radian, in mathematics, is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees. It is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written as "1.
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    degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of plane angle, representing 1⁄360 of a full rotation.
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    Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
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    A grade (or gradient) is the pitch of a slope, and is often expressed as a percent tangent, or "rise over run". It is used to express the steepness of slope on a hill, stream, roof, railroad, or road, where zero indicates level
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    Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass.

    The ruler to be used is assumed to be infinite in length, has no markings on it and only one edge, and is known as a
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    circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the centre.

    Circles are simple closed curves which divide the plane into an interior and exterior.
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    Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements is the earliest known systematic discussion of geometry.
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    In geometry, the parallel postulate, also called Euclid's fifth postulate since it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry.
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    vertical (US English) or opposite (British English) if the angles share the same vertex and are bounded by the same pair of lines but are opposite to each other. Such angles are congruent and thus have equal measure.
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    In mathematics, orthogonal, as a simple adjective, not part of a longer phrase, is a generalization of perpendicular. It means at right angles, from the Greek ὀρθός orthos
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    tangential component of the vector, and another one perpendicular to the surface, called the normal component of the vector.

    More generally, given a submanifold N of a manifold M, and a vector in the tangent space to M at a point of
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    surface normal, or simply normal, to a flat surface is a vector which is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P.
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