# Dictionary Definition

perpendicularity

### Noun

1 the relation of opposition between things at
right angles [syn: orthogonality, orthogonal
opposition]

2 the quality of being at right angles to a given
line or plane (especially the plane of the horizon)

# User Contributed Dictionary

## English

### Noun

- The condition of being perpendicular

# Extensive Definition

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.

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.

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 .

## Numerical criteria

### In terms of slopes

In a Cartesian coordinate system, two straight lines L and M may be described by equations.- L : y = ax + b,
- M : y = cx + d,

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

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.

## In relationship to parallel lines

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'.

## 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

## 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

perpendicularity in Bulgarian:
Перпендикуляр

perpendicularity in Catalan: Perpendicular

perpendicularity in Czech: Kolmice

perpendicularity in German: Orthogonalität

perpendicularity in Spanish: Perpendicular

perpendicularity in Esperanto:
Perpendikularo

perpendicularity in French:
Perpendicularité

perpendicularity in Classical Chinese: 垂直

perpendicularity in Hebrew: אנך

perpendicularity in Dutch: Loodrecht
(meetkunde)

perpendicularity in Japanese: 垂直

perpendicularity in Polish: Prostopadłość

perpendicularity in Portuguese:
Perpendicularidade

perpendicularity in Russian:
Перпендикулярность

perpendicularity in Slovenian:
Pravokotnost

perpendicularity in Finnish: Kohtisuora

perpendicularity in Swedish: Vinkelrät

perpendicularity in Chinese:
垂直