Perpendicular
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Image:Perpendicular-coloured.svg
- For other uses, see Perpendicular (disambiguation).
In geometry, two lines are considered perpendicular if one falls on the other in such a way as to create two equal angles. The term may be used as a noun or adjective. Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B.
If a line is perpendicular to another as in Figure 1, the two angles created are called right angles, or angles measuring 90°. The line AB does not have to end at B to be considered perpendicular.
Compare to parallel.
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Numerical criteria
In terms of slopes
In a Cartesian coordinate system, two straight lines <math>L</math> and <math>M</math> may be described by equations
- <math>L : y = ax + b,</math>
- <math>M : y = cx + d,</math>
as long as neither is vertical. Then <math>a</math> and <math>b</math> are the slopes of the two lines. The lines <math>L</math> and <math>M</math> are perpendicular if and only if the product of their slopes is -1, or if <math>ac=-1</math>.
In terms of direction vectors
Two vectors in the Cartesian plane are perpendicular if and only if their dot product vanishes. More precisley, the line joining the origin (0,0) to the point (a,b) and the line joining (0,0) with (c,d) are perpendicular if and only if the quantity (a,b)ˑ(c,d) = ac + bd is equal to 0.
Construction of the perpendicular
Image:Perpendicular-construction.svg To construct the perpendicular to the line AB through the point P using compass and straightedge, proceed as follows (see Figure 2).
- 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', both passing through P. Let Q be the other point of intersection of these two circles.
- Step 3 (blue): connect P and Q to construct the desired perpendicular PQ.
To prove that the PQ is perpendicular to AB, use the SSS congruence theorem for triangles QPA' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the SAS congruence theorem for triangls OPA' and OPB' to conclude that angles POA and POB are equal.
See also
de:Orthogonalität es:Perpendicular fi:Kohtisuora fr:perpendicularité he:אנך ja:垂直 nl:Loodrecht (meetkunde) ru:Перпендикулярность zh:垂直