Editing Perpendicular (section)

== Construction of the perpendicular ==
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| image1 = Perpendicular-construction.svg
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| caption1 = Construction of the perpendicular (blue) to the line AB through the point P.
| image2 = 01-Rechter Winkel mittels Thaleskreis.gif
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| caption2 =  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
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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 [[Congruence (geometry)#Congruence of triangles|SSS congruence theorem]] for QPA' and QPB' to conclude that angles OPA' and OPB' are equal.  Then use the [[Congruence (geometry)#Congruence of triangles|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.