Perpendicular

In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. 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 vector. A line is said to be perpendicular to another line if the two lines intersect at a right angle. 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 A B ¯ {\displaystyle {\overline {AB}}} is perpendicular to a line segment C D ¯ {\displaystyle {\overline {CD}}} 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, A B ¯ ⊥ C D ¯ {\displaystyle {\overline {AB}}\perp {\overline {CD}}} means line segment AB is perpendicular to line segment 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.

Foot of a perpendicular 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.

Construction of the perpendicular

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. 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

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 computing distances

Graph of functions

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