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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i In Geometry, adjacent angles are Angles that have a common ray coming out of the vertex going between two other rays The Angles is a modern English word for a Germanic-speaking people who took their name from the cultural ancestral region of Angeln, a modern district located in The term may be used as a noun or adjective. In Grammar, an adjective is a word whose main syntactic role is to modify a Noun or Pronoun, giving more information about the Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B. Note that by definition, a line is infinitely long, and strictly speaking AB and CD in this example represent line segments of two infinitely long lines. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points 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 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°). In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 This article describes the unit of angle For other meanings see Degree. Conversely, any lines that meet to form right angles are perpendicular.

In a coordinate plane, perpendicular lines have opposite reciprocal slopes. 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 ABC DEF .

Numerical criteria

In terms of slopes

In a Cartesian coordinate system, two straight lines L and M may be described by equations. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

L:y = ax + b,

M:y = cx + d,

as long as neither is vertical. Then a and c are the slopes of the two lines. The grade (or gradient or pitch or slope) of any physical feature such as a Hill, Stream, Roof, railroad, or The lines L and M are perpendicular if and only if the product of their slopes is -1, or if ac = − 1.

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 P:x = J and horizontal line Q:y = K, where J and K are constants, P Q .

Construction of the perpendicular

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). Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles

• 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. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Distance is a numerical description of how far apart objects are
• 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. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i Then use the SAS congruence theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i

In relationship to parallel lines

Fig. 3: Lines a and b are parallel, as shown by the tick marks, and are cut by the transversal line c. In Geometry, a Transversal line is a line that passes through two or more other Coplanar lines at different points.

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. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive 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. pair of Angles is said to be vertical (US English or opposite (British English if the angles share the same vertex and are bounded by the same pair of 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'.

Finding the perpendiculars of a function

Algebra

In algebra, for any linear equation y=mx + b, the perpendiculars will all have a slope of (-1/m), the opposite reciprocal of the original slope. It is helpful to memorize the slogan "to find the slope of the perpendicular line, flip the fraction and change the sign. " Recall that any whole number a is itself over one, and can be written as (a/1)

To find the perpendicular of a given line which also passes through a particular point (x, y), solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b.

Calculus

First find the derivative of the function. This will be the slope (m) of any curve at a particular point (x, y). Then, as above, solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b.

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