Three lengths of chain in the ratio 3, 4, 5 make a right angle
Gardeners and surveyors have a low-tech way to mark out a square corner across a large field: lay three chains in the ratio 3:4:5 into a triangle. The corner opposite the longest side will be a right angle. Perpendicular lines are everywhere, and so are clever ways to draw them.
Two objects are perpendicular when they meet at 90 degrees, or pi over two radians. That can mean two lines, a line and a plane, or two planes. More precisely, one line is perpendicular to another if they cross and the second line splits the straight angle on one side of the first into two equal halves. The relationship works both ways, so the order does not matter; the compass rose, with north to south crossing west to east, is a familiar picture. A line is perpendicular to a plane when it is perpendicular to every line in that plane it touches, and two planes are perpendicular when they meet at a right dihedral angle.
The point where a perpendicular from a given point lands on a line is called its foot, even when the diagram is turned so it sits at the top. With just a compass and straightedge, you can drop one from a point: draw a circle that cuts the line twice, draw two equal circles from those crossing points, and join the places where they meet. Congruent triangles prove the result is square.
Perpendicularity ties closely to parallels. In Euclidean geometry, two lines both perpendicular to a third must be parallel, thanks to the parallel postulate, and a line perpendicular to one of two parallels is perpendicular to the other. In coordinates, two lines are perpendicular when the product of their slopes is minus one, and vectors are perpendicular when their dot product vanishes. Mathematicians treat the whole idea as the classical case of the broader notion of orthogonality.
Circles are full of right angles. A diameter meets the tangent at its endpoint squarely, and a line from the centre that bisects a chord is perpendicular to it. Thales' theorem says a diameter makes a right angle at every other point on the circle. And if two perpendicular chords cross, the squares of their four pieces add up to the square of the diameter.
Source: Perpendicular
