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In elementary geometry, two geometric objects are **perpendicular** if their intersection forms right angles (angles that are 90 degrees or π/2 radians wide) at the point of intersection called a *foot*. 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.

- Foot of a perpendicular
- Construction of the perpendicular
- In relationship to parallel lines
- In computing distances
- Graph of functions
- In circles and other conics
- Circles
- Ellipses
- Parabolas
- Hyperbolas
- In polygons
- Triangles
- Quadrilaterals
- Lines in three dimensions
- See also
- Notes
- References
- External links

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 vector *.

A line is said to be perpendicular to another line if the two lines intersect at a right angle.^{ [2] } 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 congruent angles. 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 segments and rays. For example, a line segment is perpendicular to a line segment 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, means line segment AB is perpendicular to line segment CD.^{ [3] }

A line is said to be perpendicular to a 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.

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 A be a point and m a line. If B is the point of intersection of m and the unique line through A that is perpendicular to m, then B is called the *foot* of this perpendicular through A.

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 SSS congruence theorem for 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.

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.

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 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 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 geometry, the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both.

The distance from a point to a line is the distance to the nearest point on that line. That is the point at which a segment from it to the given point is perpendicular to the line.

Likewise, the distance from a point to a curve is measured by a line segment that is perpendicular to a tangent line to the curve at the nearest point on the curve.

The distance from a point to a plane is measured as the length from the point along a segment that is perpendicular to the plane, meaning that it is perpendicular to all lines in the plane that pass through the nearest point in the plane to the given point.

Other instances include:

*Point on plane closest to origin*, for the perpendicular distance from the origin to a plane in three-dimensional space*Nearest distance between skew lines*, for the perpendicular distance between two non-parallel lines in three-dimensional space

Perpendicular regression fits a line to data points by minimizing the sum of squared perpendicular distances from the data points to the line. Other geometric curve fitting methods using perpendicular distance to measure the quality of a fit exist, as in total least squares.

The concept of perpendicular distance may be generalized to

- orthogonal distance, between more abstract non-geometric orthogonal objects, as in linear algebra (e.g., principal components analysis);
- normal distance, involving a surface normal, between an arbitrary point and its foot on the surface. It can be used for surface fitting and for defining offset surfaces.

In the two-dimensional plane, right angles can be formed by two intersected lines if the product of their slopes equals −1. Thus defining two linear functions: *y*_{1} = *a*_{1}*x* + *b*_{1} and *y*_{2} = *a*_{2}*x* + *b*_{2}, the graphs of the functions will be perpendicular and will make four right angles where the lines intersect if *a*_{1}*a*_{2} = −1. However, this method cannot be used if the slope is zero or undefined (the line is parallel to an axis).

For another method, let the two linear functions be: *a*_{1}*x* + *b*_{1}*y* + *c*_{1} = 0 and *a*_{2}*x* + *b*_{2}*y* + *c*_{2} = 0. The lines will be perpendicular if and only if *a*_{1}*a*_{2} + *b*_{1}*b*_{2} = 0. This method is simplified from the dot product (or, more generally, the inner product) of vectors. In particular, two vectors are considered orthogonal if their inner product is zero.

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 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 *a*^{2} + *b*^{2} + *c*^{2} + *d*^{2} equals the square of the diameter.^{ [4] }

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*^{2} – 4*p*^{2} (where *r* is the circle's radius and *p* is the distance from the center point to the point of intersection).^{ [5] }

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.

The major and minor 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 and to each latus rectum.

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 other tangent line to the parabola is perpendicular to the line from that point through the parabola's focus.

The 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.

The 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 rectangular hyperbola has asymptotes that are perpendicular to each other. It has an eccentricity equal to

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

The altitudes of a triangle are perpendicular to their respective bases. The perpendicular bisectors 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 and perpendicular to any line tangent to the triangle's incircle.

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 maltitudes of a quadrilateral is a perpendicular to a side through the midpoint of the opposite side.

An orthodiagonal quadrilateral is a quadrilateral whose diagonals are perpendicular. These include the square, the rhombus, and the kite. By Brahmagupta's theorem, in an orthodiagonal quadrilateral that is also 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.

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.

A **circle** is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

In mathematics, a **parabola** is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

In geometry, **bisection** is the division of something into two equal or congruent parts. Usually it involves a bisecting line, also called a *bisector*. The most often considered types of bisectors are the *segment bisector* and the *angle bisector*.

In Euclidean geometry, a **cyclic quadrilateral** or **inscribed quadrilateral** is a quadrilateral whose vertices all lie on a single circle. This circle is called the *circumcircle* or *circumscribed circle*, and the vertices are said to be *concyclic*. The center of the circle and its radius are called the *circumcenter* and the *circumradius* respectively. Other names for these quadrilaterals are **concyclic quadrilateral** and **chordal quadrilateral**, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

In mathematics, **hyperbolic geometry** is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

In geometry, the **midpoint** is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

In geometry, a set of points are said to be **concyclic** if they lie on a common circle. All concyclic points are equidistant from the center of the circle. Three points in the plane that do not all fall on a straight line are concyclic, but four or more such points in the plane are not necessarily concyclic.

In geometry, the **Dandelin spheres** are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres are also sometimes called **focal spheres**.

In hyperbolic geometry, two lines are said to be **ultraparallel** if they do not intersect and are not limiting parallel.

A point is said to be **equidistant** from a set of objects if the distances between that point and each object in the set are equal.

In geometry, lines in a plane or higher-dimensional space are **concurrent** if they intersect at a single point. They are in contrast to parallel lines.

In hyperbolic geometry, an **ideal point**, **omega point** or **point at infinity** is a well-defined point outside the hyperbolic plane or space. Given a line *l* and a point *P* not on *l*, right- and left-limiting parallels to *l* through *P* converge to *l* at *ideal points*.

In hyperbolic geometry, a **hypercycle**, **hypercircle** or **equidistant curve** is a curve whose points have the same orthogonal distance from a given straight line.

A **conic section**, **conic** or a **quadratic curve** is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.

The **intersecting chords theorem** or just the **chord theorem** is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's *Elements*.

In Euclidean plane geometry, a **tangent line to a circle** is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.

In Euclidean geometry, an **orthodiagonal quadrilateral** is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

In plane geometry, an **extended side** or **sideline** of a polygon is the line that contains one side of the polygon. The extension of a finite side into an infinite line arises in various contexts.

Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed. The fifth axiom of hyperbolic geometry says that given a line *L* and a point *P* not on that line, there are at least two lines passing through *P* that are parallel to *L*. As in Euclidean geometry, where ancient Greek mathematicians used a compass and idealized ruler for constructions of lengths, angles, and other geometric figures, constructions can also be made in hyperbolic geometry.

In geometry, the **Newton–Gauss line** is the line joining the midpoints of the three diagonals of a complete quadrilateral.

- Altshiller-Court, Nathan (1925),
*College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle*(2nd ed.), New York: Barnes & Noble, LCCN 52-13504 - Kay, David C. (1969),
*College Geometry*, New York: Holt, Rinehart and Winston, LCCN 69-12075

Look up ** perpendicular ** in Wiktionary, the free dictionary.

- 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).

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Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.