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 calledthe perpendicular from A to the segment CD, using "perpendicular" as a noun. The pointB is called thefoot of the perpendicular fromA to segment CD, or simply, thefoot ofA on CD.[1]
Ingeometry, twogeometric objects areperpendicular if theyintersect atright angles, i.e. at anangle of 90 degrees or π/2 radians. The condition ofperpendicularity may be represented graphically using theperpendicular 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 oforthogonality; 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 itsnormal 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 thestraight angle on one side of the first line is cut by the second line into twocongruentangles. Perpendicularity can be shown to besymmetric, 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 tosegments andrays. 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 aplane 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 thedihedral angle at which they meet is a right angle.
The wordfoot 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, letA be a point andm a line. IfB is the point of intersection ofm and the unique line throughA that is perpendicular tom, thenB is called thefoot of this perpendicular throughA.
Construction of the perpendicular (blue) to the line AB through the point P
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
Step 1 (red): construct acircle with center at P to create points A' and B' on the line AB, which areequidistant 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 theSSS congruence theorem for QPA' and QPB' to conclude that angles OPA' and OPB' are equal. Then use theSAS congruence theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal. See alsoRadical axis.
To make the perpendicular to the line g at or through the point P usingThales's theorem, see the animation at right.
ThePythagorean 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.
The arrowhead marks indicate that the linesa andb, cut by thetransversal linec, are parallel.
If two lines (a andb) are both perpendicular to a third line (c), all of the angles formed along the third line are right angles. Therefore, inEuclidean geometry, any two lines that are both perpendicular to a third line areparallel to each other, because of theparallel 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, becausevertical angles are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if linesa andb 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.
Thedistance from a point to a line is the distance to the nearestpoint on that line. That is the point at which asegment from it to the given point is perpendicular to the line.
Likewise, the distance from a point to acurve is measured by a line segment that is perpendicular to atangent line to the curve at the nearest point on the curve.
Thedistance 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.
Perpendicular regression fits a line to data points by minimizing the sum of squared perpendicular distances from the data points to the line.Othergeometric curve fitting methods using perpendicular distance to measure the quality of a fit exist, as intotal least squares.
The concept of perpendicular distance may be generalized to
Two perpendicular lines have slopesm1 = Δy1/Δx1 andm2 = Δy2/Δx2 satisfying the relationshipm1m2 = −1.
In the two-dimensional plane, right angles can be formed by two intersected lines if theproduct of theirslopes equals −1. Thus for twolinear functions and, the graphs of the functions will be perpendicular if
Thedot product ofvectors can be also used to obtain the same result: First,shift coordinates so that the origin is situated where the lines cross. Then define two displacements along each line,, for Now, use the fact that the inner product vanishes for perpendicular vectors:
(unless or vanishes.)
Both proofs are valid for horizontal and vertical lines to the extent that we can let one slope be, and take the limit that If one slope goes to zero, the other goes to infinity.
Eachdiameter of acircle is perpendicular to thetangent line to that circle at the point where the diameter intersects the circle.
A line segment through a circle's center bisecting achord is perpendicular to the chord.
If the intersection of any two perpendicular chords divides one chord into lengthsa andb and divides the other chord into lengthsc andd, thena2 +b2 +c2 +d2 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 8r2 – 4p2 (wherer is the circle's radius andp 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 minoraxes of anellipse 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 thedirectrix and to eachlatus rectum.
In aparabola, 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, theother tangent line to the parabola is perpendicular to the line from that point through the parabola'sfocus.
Theorthoptic 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 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.
In asquare or otherrectangle, all pairs of adjacent sides are perpendicular. Aright trapezoid is atrapezoid that has two pairs of adjacent sides that are perpendicular.
Anorthodiagonal quadrilateral is a quadrilateral whosediagonals are perpendicular. These include thesquare, therhombus, and thekite. ByBrahmagupta's theorem, in an orthodiagonal quadrilateral that is alsocyclic, a line through the midpoint of one side and through the intersection point of the diagonals is perpendicular to the opposite side.
Byvan 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.
