Perpendicular

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The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees.
For other uses, see Perpendicular (disambiguation).

In elementary geometry, perpendicularity or the property of being perpendicular is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects as well.

A line is said to be perpendicular to another line if the two lines intersect at a right angle.[1] 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.

Perpendicularity easily extends to segments and rays. For example, we say a line segment \overline{AB} is perpendicular to a line segment \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, we write \overline{AB} \perp \overline{CD} to mean line segment AB is perpendicular to line segment CD.[2] The point B is called a foot of the perpendicular from A to segment \overline{CD}, or simply, a foot of A on \overline{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. Note that 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 degrees).

In fact, 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.

Construction of the perpendicular[edit]

Construction of the perpendicular (blue) to the line AB through the point P.

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

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

To prove that the 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.

In a Cartesian coordinate system[edit]

In a Cartesian coordinate system, if two lines are each not parallel to either of the coordinate axes, then the two lines are perpendicular if and only if the product of their gradients is −1.

In relationship to parallel lines[edit]

Lines a and b are parallel, as shown by the tick marks, and are cut by the transversal line c.

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'.

Graph of functions[edit]

In the 2-dimensional plane, right angles can be formed by two intersected lines which the product of their slopes equals to −1. More precisely, defining two linear functions: y1 = a1x + b1 and y2 = a2x + b2, the graph of the functions will be perpendicular and will make four right angles where the lines intersect if and only if a1a2 = −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: a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0. The lines will be perpendicular if and only if a1a2 + b1b2 = 0. This method is simplified from the dot product (or generally, inner product) of vectors. In particular, two vectors are considered orthogonal if their inner product is zero.

See also[edit]

Notes[edit]

  1. ^ Kay (1969, p. 91)
  2. ^ Kay (1969, p. 91)
  3. ^ Kay (1969, p. 114)

References[edit]

External links[edit]