Editing Perpendicular

{{short description|Relationship between two lines that meet at a right angle}}
{{other uses}}
[[File:Perpendicular-coloured.svg|class=skin-invert-image|right|thumb|236px<!--(approx Sidebar/Infobox)-->|The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. The segment AB can be called ''the perpendicular from A to the segment CD'', using "perpendicular" as a noun. The point ''B'' is called the ''foot of the perpendicular from ''A'' to segment CD'', or simply, the ''foot of ''A'' on CD''.<ref>{{harvtxt|Kay|1969|p=114}}</ref> ]]
{{General geometry |concepts}}

In [[geometry]], two [[geometric object]]s are '''perpendicular''' if they [[intersection|intersect]] at [[right angle|right angles]], i.e. at an [[angle]] of 90 degrees or π/2 radians. <!-- Do we really need to define a right angle to someone who knows what radians are? --> The condition of '''perpendicularity''' may be represented graphically using the ''[[perpendicular symbol]]'', ⟂. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes.

''Perpendicular'' is also used as a noun: '''a perpendicular''' is a line which is perpendicular to a given line or plane.

Perpendicularity is one particular instance of the more general mathematical concept of ''[[orthogonality]]''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its ''[[Normal (geometry)|normal vector]]''.

A line is said to be perpendicular to another line if the two lines intersect at a right angle.<ref>{{harvtxt|Kay|1969|p=91}}</ref>  Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the [[straight angle]] on one side of the first line is cut by the second line into two [[Congruence (geometry)|congruent]] [[angle]]s. Perpendicularity can be shown to be [[symmetric]], meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order. A great example of perpendicularity can be seen in any compass, note the cardinal points; North, East, South, West (NESW)
The line N-S is perpendicular to the line W-E and the angles N-E, E-S, S-W and W-N are all 90° to one another.

Perpendicularity easily extends to [[Line segment|segment]]s and [[Ray (geometry)|ray]]s. For example, a line segment <math>\overline{AB}</math> is perpendicular to a line segment <math>\overline{CD}</math> if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols, <math>\overline{AB} \perp \overline{CD}</math> means line segment AB is perpendicular to line segment CD.<ref>{{harvtxt|Kay|1969|p=91}}</ref>

A line is said to be perpendicular to a [[Plane (geometry)|plane]] if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines.

Two planes in space are said to be perpendicular if the [[dihedral angle]] at which they meet is a right angle.

==Foot of a perpendicular{{anchor|Foot}}==
The word '''foot''' is frequently used in connection with perpendiculars. This usage is exemplified in the top diagram, above, and its caption. The diagram can be in any orientation. The foot is not necessarily at the bottom.

More precisely, let {{mvar|A}} be a point and {{mvar|m}} a line. If {{mvar|B}} is the point of intersection of {{mvar|m}} and the unique line through {{mvar|A}} that is perpendicular to {{mvar|m}}, then {{mvar|B}} is called the ''foot'' of this perpendicular through {{mvar|A}}.
{{clear}}

== Construction of the perpendicular ==
<div class='skin-invert-image'>{{multiple image
| align = right
| image1 = Perpendicular-construction.svg
| width1 = 236 
| alt1 = 
| caption1 = Construction of the perpendicular (blue) to the line AB through the point P.
| image2 = 01-Rechter Winkel mittels Thaleskreis.gif
| width2 = 256
| alt2 = 
| caption2 =  Construction of the perpendicular to the half-line h from the point P (applicable not only at the end point A, M is freely selectable), animation at the end with pause 10 s
| footer = 
}}</div>

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

* 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 P be the points of intersection of these two circles.
* Step 3 (blue): connect Q and P to construct the desired perpendicular PQ.

To prove that the PQ is perpendicular to AB, use the [[Congruence (geometry)#Congruence of triangles|SSS congruence theorem]] for QPA' and QPB' to conclude that angles OPA' and OPB' are equal.  Then use the [[Congruence (geometry)#Congruence of triangles|SAS congruence theorem]] for triangles OPA' and OPB' to conclude that angles POA and POB are equal. See also [[Radical axis]].

To make the perpendicular to the line g at or through the point P using [[Thales's theorem]], see the animation at right.

The [[Pythagorean theorem]] can be used as the basis of methods of constructing right angles. For example, by counting links, three pieces of chain can be made with lengths in the ratio 3:4:5. These can be laid out to form a triangle, which will have a right angle opposite its longest side. This method is useful for laying out gardens and fields, where the dimensions are large, and great accuracy is not needed. The chains can be used repeatedly whenever required.

== In relationship to parallel lines ==
[[File:perpendicular transversal v3.svg|class=skin-invert-image|thumb|236px<!--(as above)-->|The arrowhead marks indicate that the lines ''a'' and ''b'', cut by the [[transversal line]] ''c'', are parallel.]]

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 (geometry)|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)|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''.
* All four angles are equal.

==In computing distances{{anchor|Distance}}==
{{excerpt|Perpendicular distance}}

== Graph of functions ==
[[File:Slopes and orthogonality.svg|thumb|upright=.95|Two perpendicular lines have slopes {{math|1=''m''<sub>1</sub> = Δ''y''<sub>1</sub>/Δ''x''<sub>1</sub>}} and  {{math|1=''m''<sub>2</sub> = Δ''y''<sub>2</sub>/Δ''x''<sub>2</sub>}} satisfying the relationship  {{math|1=''m''<sub>1</sub>''m''<sub>2</sub> = &minus;1}}.]]
In the two-dimensional plane, right angles can be formed by two intersected lines if the [[Product (mathematics)|product]] of their [[slopes]] equals  −1. Thus for two [[linear function]]s <math>y_1(x) = m_1 x + b_1</math> and <math>y_2(x) = m_2 x + b_2</math>, the graphs of the functions will be perpendicular if <math>m_1 m_2 = -1.</math> 

The [[dot product]] of [[Euclidean vector|vector]]s can be also used to obtain the same result: First, [[Translation of axes|shift coordinates]] so that the origin is situated where the lines cross. Then define two displacements along each line, <math>\vec r_j</math>, for <math>(j=1,2).</math> Now, use the fact that the inner product vanishes for perpendicular vectors:
:<math>\vec r_1=x_1\hat x + y_1\hat y =x_1\hat x + m_1x_1\hat y</math>
:<math>\vec r_2=x_2\hat x + y_2\hat y = x_2\hat x + m_2x_2\hat y</math>
:<math>\vec r_1 \cdot \vec r_2 = \left(1+m_1m_2\right)x_1x_2 =0</math>
:<math>\therefore m_1m_2=-1</math> (unless  <math>x_1</math> or <math>x_2</math> vanishes.)

Both proofs are valid for horizontal and vertical lines to the extent that we can let one slope be <math>\varepsilon</math>, and take the limit that <math>\varepsilon\rightarrow 0.</math>  If one slope goes to zero, the other goes to infinity.

==In circles and other conics==

===Circles===

Each [[diameter]] of a [[circle]] is perpendicular to the [[tangent line]] to that circle at the point where the diameter intersects the circle.

A line segment through a circle's center bisecting a [[chord (geometry)|chord]] is perpendicular to the chord.

If the intersection of any two perpendicular chords divides one chord into lengths ''a'' and ''b'' and divides the other chord into lengths ''c'' and ''d'', then {{nowrap|''a''<sup>2</sup> + ''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup>}} equals the square of the diameter.<ref>Posamentier and Salkind, ''Challenging Problems in Geometry'', Dover, 2nd edition, 1996: pp. 104–105, #4–23.</ref>

The sum of the squared lengths of any two perpendicular chords intersecting at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8''r''<sup>2</sup> – 4''p''<sup>2</sup> (where ''r'' is the circle's radius and ''p'' is the distance from the center point to the point of intersection).<ref>''[[College Mathematics Journal]]'' 29(4), September 1998, p. 331, problem 635.</ref>

[[Thales' theorem]] states that two lines both through the same point on a circle but going through opposite endpoints of a diameter are perpendicular. This is equivalent to saying that any diameter of a circle subtends a right angle at any point on the circle, except the two endpoints of the diameter.

===Ellipses===

The major and minor [[axis of symmetry|axes]] of an [[ellipse]] are perpendicular to each other and to the tangent lines to the ellipse at the points where the axes intersect the ellipse.

The major axis of an ellipse is perpendicular to the [[directrix (conic section)|directrix]] and to each [[latus rectum]].

===Parabolas===

In a [[parabola]], the axis of symmetry is perpendicular to each of the latus rectum, the directrix, and the tangent line at the point where the axis intersects the parabola.

From a point on the tangent line to a parabola's vertex, the [[Parabola#Intersection of a tangent and perpendicular from focus|other tangent line to the parabola]] is perpendicular to the line from that point through the parabola's [[focus (geometry)|focus]].

The [[Parabola#Orthoptic property|orthoptic property]] of a parabola is that If two tangents to the parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular. This implies that, seen from any point on its directrix, any parabola subtends a right angle.

===Hyperbolas===

The [[hyperbola#Equation|transverse axis]] of a [[hyperbola]] is perpendicular to the conjugate axis and to each directrix.

The product of the perpendicular distances from a point P on a hyperbola or on its conjugate hyperbola to the asymptotes is a constant independent of the location of P.

A [[hyperbola#Rectangular hyperbola|rectangular hyperbola]] has [[asymptote]]s that are perpendicular to each other. It has an [[eccentricity (mathematics)|eccentricity]] equal to <math>\sqrt{2}.</math>

==In polygons==

===Triangles===

The legs of a [[right triangle]] are perpendicular to each other.

The [[altitude (geometry)|altitudes]] of a [[triangle]] are perpendicular to their respective [[base (geometry)|bases]]. The [[perpendicular bisector]]s of the sides also play a prominent role in triangle geometry.

The [[Euler line]] of an [[isosceles triangle]] is perpendicular to the triangle's base.

The [[Droz-Farny line theorem]] concerns a property of two perpendicular lines intersecting at a triangle's [[orthocenter]].

[[Harcourt's theorem]] concerns the relationship of line segments through a [[vertex (geometry)|vertex]] and perpendicular to any line [[tangent]] to the triangle's [[incircle]].

===Quadrilaterals===

In a [[square]] or other [[rectangle]], all pairs of adjacent sides are perpendicular. A [[right trapezoid]] is a [[trapezoid]] that has two pairs of adjacent sides that are perpendicular.

Each of the four [[maltitude]]s of a [[quadrilateral]] is a perpendicular to a side through the [[midpoint]] of the opposite side.

An [[orthodiagonal quadrilateral]] is a quadrilateral whose [[diagonal]]s are perpendicular. These include the [[square]], the [[rhombus]], and the [[kite (geometry)|kite]]. By [[Brahmagupta's theorem]], in an orthodiagonal quadrilateral that is also [[cyclic quadrilateral|cyclic]], a line through the midpoint of one side and through the intersection point of the diagonals is perpendicular to the opposite side.

By [[van Aubel's theorem]], if squares are constructed externally on the sides of a quadrilateral, the line segments connecting the centers of opposite squares are perpendicular and equal in length.

==Lines in three dimensions==

Up to three lines in [[three-dimensional space]] can be pairwise perpendicular, as exemplified by the ''x, y'', and ''z'' axes of a three-dimensional [[Cartesian coordinate system]].

== See also ==
* [[Orthogonal projection]]
* [[Tangential and normal components]]

== Notes ==
{{reflist}}

== References ==
* {{citation |last=Altshiller-Court |first=Nathan |author-link=Nathan Altshiller Court |year=1952 |orig-year=1st ed. 1925 |title=College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle |location=New York |publisher=Barnes & Noble |edition=2nd |url=https://archive.org/details/collegegeometryi00newy/ |url-access=limited }}
* {{citation |first1=David C. |last1=Kay |year=1969 |lccn=69-12075 |title=College Geometry |publisher=[[Holt, Rinehart and Winston]] |location=New York}}

== External links ==
{{Wiktionary}}
*[http://www.mathopenref.com/perpendicular.html Definition: perpendicular] with interactive animation.
*[http://www.mathopenref.com/constbisectline.html How to draw a perpendicular bisector of a line with compass and straight edge] (animated demonstration).
*[http://www.mathopenref.com/constperpendray.html How to draw a perpendicular at the endpoint of a ray with compass and straight edge] (animated demonstration).

[[Category:Orthogonality]]