Perpendicular

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.

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.

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

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.

Therefore, if lines a and b are parallel, any of the following conclusions leads to all of the others: In geometry, the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both.

Other instances include: Perpendicular regression fits a line to data points by minimizing the sum of squared perpendicular distances from the data points to the line.

The concept of perpendicular distance may be generalized to In the two-dimensional plane, right angles can be formed by two intersected lines if the product of their slopes equals −1.

The dot product of vectors can be also used to obtain the same result: First, shift coordinates so that the origin is situated where the lines cross.

Now, use the fact that the inner product vanishes for perpendicular vectors: Both proofs are valid for horizontal and vertical lines to the extent that we can let one slope be

[5] Thales' theorem states that two lines both through the same point on a circle but going through opposite endpoints of a diameter are perpendicular.

The major axis of an ellipse is perpendicular to the directrix and to each latus rectum.

In a parabola, the axis of symmetry is perpendicular to each of the latus rectum, the directrix, and the tangent line at the point where the axis intersects the parabola.

Conversely, two tangents which intersect on the directrix are perpendicular.

This implies that, seen from any point on its directrix, any parabola subtends a right angle.

The altitudes of a triangle are perpendicular to their respective bases.

The perpendicular bisectors of the sides also play a prominent role in triangle geometry.

In a square or other rectangle, all pairs of adjacent sides are perpendicular.

By Brahmagupta's theorem, in an orthodiagonal quadrilateral that is also cyclic, a line through the midpoint of one side and through the intersection point of the diagonals is perpendicular to the opposite side.

By van Aubel's theorem, if squares are constructed externally on the sides of a quadrilateral, the line segments connecting the centers of opposite squares are perpendicular and equal in length.