# Perpendicular Lines

In elementary geometry, the property of being perpendicular is the relation between two lines that meet at a right angle (90 degrees ). The property extends to other related geometric objects .

Segment AB is perpendicular to segment CD because the two angles (shown in orange and blue) subtended by it are 90 degrees each. The segment AB can be said to be perpendicular to the segment CD from A, by using “perpendicular” as a noun . Point B is called the leg of a segment perpendicular to the CD , or simply, the leg of a CD .

A line is said to be perpendicular to another line if two lines intersect at a right angle . [2] Clearly, the first line is perpendicular to the second line if (1) the two lines meet; and (2) at the point of intersection a straight angle on one side of the first line is cut by the second line into two congruent angles . The perpendicularity can be shown to be symmetric , which means that if the first line is perpendicular to the second, then the second row is also perpendicular to the first. For this reason, we can speak of two lines being perpendicular (to each other) without specifying a sequence.

The 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, i.e. line segment AB is perpendicular to line segment CD. See spike above for information about the vertical symbol .

{\overline {AB}}\perp {\overline {CD}}

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 (90°).

Perpendicularity is a special example of the more general mathematical concept of orthogonality ; Perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the term “perpendicular” is sometimes used to describe much more complex geometric orthogonality situations, such as between a surface and its normal .

## Leg of a Vertical

The term foot is often used in relation to the vertical. An example of this use is given in the vertex diagram, above, and in its title. The diagram can be in any orientation. It is not necessary that the foot should be down.

More precisely, suppose that A is a point and m is a line. If B is the point of intersection of m and a unique line passing through A which is perpendicular to m , then B is said to be the foot of this perpendicular through A.

## Vertical Construction

To make a line perpendicular to AB through point P using the compass-and-perpendicular construction , proceed as follows (see image at left):

• Step 1 (red): Construct a circle with center at P to make points A’ and B’ on line AB , which are equidistant from P.
• Step 2 (Green): Draw circles centered at A’ and B’ of equal radii. Let Q and P be the points of intersection of these two circles.
• Step 3 (Blue): Connect Q and P to form the desired vertical PQ.

To prove that PQ is perpendicular to AB, use the SSS congruence theorem for ‘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 draw a perpendicular to the line g or through point P using Thales’ theorem , see the animation on the right.

The Pythagorean theorem can be used as a basis for methods for constructing right angles. For example, by counting the 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 whose longest side will have a right angle opposite it. This method is useful for laying gardens and fields, where the dimensions are large, and great accuracy is not required. The chains can be used over and over again whenever needed.

## With Respect to Parallel Lines

If two lines ( A and B ) are both perpendicular to the third line ( C ), then all angles formed with the third line are right angles. Therefore, in Euclidean geometry , any two lines that are both perpendicular to the third line are parallel to each other, because of the parallel concept . Conversely, if a line is perpendicular to another line, then it is also perpendicular to any line parallel to that other line.

In the figure on the right, all orange-shaded angles are congruent to each other and all green-shaded angles are congruent to each other, because vertical angles are congruent and alternate interior angles formed by parallel lines intersecting a transverse Therefore, if the lines a and b are parallel, then any of the following conclusions leads to all others:

• One of the angles in the diagram is a right angle.
• One of the orange-shaded angles corresponds to one of the green-shaded angles.
• Line c is perpendicular to line a .
• Line c is perpendicular to line b.

### Computing Distance

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

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

Vertical regression fits the data points by minimizing the sum of the squared perpendicular distances from the data points to the line.

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

In a two-dimensional plane, two intersecting lines can form a right angle if the product of their slopes is equal to -1. Defining two linear functions as follows : y 1 = a 1 x + b 1 and y 2 = a 2 x + b 2 , the graphs of the function will be perpendicular and form 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, suppose there are two linear functions: 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 called the dot product (or, more commonly, the inner product ) of vectors .) is simplified. In particular, two vectors are said to be orthogonal if their inner product is zero.

### In Circles and Other Cones

#### Circles

Every diameter of a circle is perpendicular to the tangent to that circle at the point where the diameter intersects the circle.

A line segment passing through the center of the circle bisects the chord which is perpendicular to the chord.

If the intersection of any two perpendicular chords divides one chord into lengths a and b and another chord of lengths c and d , then a 2 + b 2 + c 2 + d 2 is equal to the square of the diameter .

The sum of the square lengths of any two perpendicular chords intersecting at a given point is equal to the sum of the other two perpendicular chords intersecting at the same point, and 8 2 – 4 2 (where r is the radius of the circle and p is the center is the distance from the point of intersection to the point of intersection).

Thales’ theorem states that two lines pass through the same point on a circle but through opposite end points of a diameter. This is equivalent to saying that any diameter of a circle subtends a right angle at any point on the circle except at the two end points of the diameter.

### Multiple Points

The major and minor axes of an ellipse are perpendicular to each other and the tangent lines of 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.

### Parabola

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

From a point on the tangent line to the vertex of a parabola, the other tangent line to the parabola is perpendicular to the line from that point through the focus of the parabola.

Orthoptic property of a parabola is that the two tangents to the parabola are perpendicular to each other, so they intersect at the directrix. In contrast, two tangents that intersect at the directrix are perpendicular. This means that, when viewed from any point in its direction, any parabola subtends a right angle.

### Hyperbola

The superlative of the transverse axis one stands for the conjugate axis and one for each coordinate.

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

A rectangular hyperbola has tangents that are perpendicular to each other. One of its singularities is equal to

{\sqrt {2}}.

### In Polygons

#### Triangle

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

The height of a triangle is perpendicular to their respective bases. The perpendicular bisectors of the sides also play a major role in triangle geometry.

The Euler line of an isosceles triangle is perpendicular to the base of the triangle.

The Droz-Farny line theorem relates the property of two perpendicular lines intersecting at the orthocenter of a triangle.

Harcourt’s theorem deals with the relation of line segments passing through a vertex and perpendicular to any line tangent to the incircle of a triangle.

In a square or other rectangle, all pairs of adjacent sides are perpendicular. A right trapezoid is a trapezoid in which two pairs of adjacent sides are perpendicular.

Each of the four vertices of a quadrilateral is perpendicular to a side passing through the midpoint of the opposite side.

An orthodiagonal quadrilateral is a quadrilateral whose diagonals are perpendicular. These include squares, rhombus and kites. By Brahmagupta’s theorem, in an orthodiagonal quadrilateral that is also cyclic, a line through the middle 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

Three lines can be pairwise perpendiculars in three-dimensional space, as exemplified by the x, y and z axes of the three-dimensional Cartesian coordinate system.