 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. 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. 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. Image File history File links Perpendicular-coloured. ...
Image File history File links Perpendicular-coloured. ...
Perpendicular may refer to: Perpendicular, in mathematics two lines at right angles Perpendicular Period, a style and period of mediaeval English Gothic architecture Perpendicular recording, a disc drive technology Purpendicular, an album by Deep Purple Category: ...
For other uses, see Geometry (disambiguation). ...
Line redirects here. ...
This article is about the mathematical construct. ...
An example of congruence. ...
In the illustration, angles A and B are adjacent. ...
White cliffs of Dover in England White cliffs of Rugen down the Baltic coast from Schleswig 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, Germany. ...
In linguistics, a noun or noun substantive is a lexical category which is defined in terms of how its members combine with other grammatical kinds of expressions. ...
In grammar, an adjective is a word whose main syntactic role is to modify a noun or pronoun (called the adjectives subject), giving more information about what the noun or pronoun refers to. ...
Line redirects here. ...
The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ...
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. This article is about angles in geometry. ...
When a circles diameter is 1, its circumference is π. Pi or π is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ...
For the musical group, see Radian (band). ...
This article describes the unit of angle. ...
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.
Numerical criteria
In terms of slopes In a Cartesian coordinate system, two straight lines L and M may be described by equations. Fig. ...
- 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 lines L and M are perpendicular if and only if the product of their slopes is -1, or if ac = − 1. In mathematics, the slope (or gradient, especially where three or more dimensions are discussed) of a straight line (within a Cartesian coordinate system) is a measure for the steepness of said line. ...
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,  .
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). Image File history File links Perpendicular-construction. ...
Image File history File links Perpendicular-construction. ...
Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ...
- 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. This article is about the shape and mathematical concept of circle. ...
personal space, proxemics. ...
An example of congruence. ...
An example of congruence. ...
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. 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. Image File history File links No higher resolution available. ...
Image File history File links No higher resolution available. ...
Transversal t cuts two parallel lines, a and b. ...
Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ...
a and b are parallel, the transversal t produces congruent angles. ...
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: Two lines intersect to create two pairs of vertical angles. ...
- 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) Look up reciprocal in Wiktionary, the free dictionary. ...
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 In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...
Illustration of tangential and normal components of a vector to a surface. ...
A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...
Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ...
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