Right angle

A right angle, and a short right, is an angle of 90 ° and therefore, the fourth part of a solid angle to 360 °. Two straight lines or lines that intersect or touch each other at a right angle are called right angles, perpendicular or orthogonal. Right angles occur in many geometric figures and designs and are identified in drawings by a small quarter circle with a dot or by a small square. The right angle was next to the full angle at times a legal unit in Germany and in Switzerland.

Definition

Both Euclid in his work The Elements (300 BC ), and David Hilbert in his system of axioms of Euclidean geometry (1899 ) define a right-angle as an angle that is congruent to its supplementary angle:

"If a straight line placed on a straight line, forming mutually equal adjacent angles, then each of the same side angle is a Right "

The adjective "right" thinks this is not right, but rather in the sense of upright (Latin rectus). Alternatively, at least since the 16th century, a right angle, is defined as an angle to the fourth circle belongs. Both definitions are equivalent to each other, because two adjacent angles that together form a straight angle, which corresponds to a semicircle.

Examples

In the plane for example, form a right angle:

• The coordinate axes of a Cartesian coordinate system,
• Two adjacent sides of a rectangle
• The two legs of a right triangle
• The two diagonals of a quadrilateral kite
• The two semi-axes of an ellipse
• The lines connecting a point on a semicircle with endpoints of the diameter ( theorem of Thales )

In space form, for example a right angle:

• Each two of the coordinate axes of a three-dimensional Cartesian coordinate system,
• Two adjacent edges of a cuboid
• Each two of the three spatial diagonal of an octahedron
• Two mutually orthogonal vectors
• The solder on a level with each straight line of the plane through the nadir point

In an orthogonal polygon or an orthogonal polyhedron all adjacent edges form right angles.

Determination of right angle

Intermediate traffic

Two routes and form according to the Pythagorean theorem if and only a right angle when the lengths of the routes

Applies. The integer solutions of this equation are called Pythagorean triples. To form two lines which meet at a point and be their lengths or units if and only each other at right angles, if the link of the two endpoints units is long because

Intermediate function graph

The graphs of two linear functions and intersect at a right angle if and only if the product of the slopes

Applies. For example, intersect the graphs of two linear functions and orthogonal, because

General intersect the graphs of two differentiable functions and exactly then at a right angle when the intersection is the product of the derivatives ( gradients of the tangent )

Results. So, for example, the graphs of the functions and the point intersect at right angles, and because

Intermediate curves

Two intersecting lines form in a Cartesian coordinate system a right angle if and only if the dot product of the direction vectors and the two lines

Applies. For example, there are two lines with direction vectors and perpendicular to each other, since

Is. General form two intersecting differentiable curves with each other at right angles, if the dot product of their tangent vectors vanishes at the intersection.

Trigonometry

For the trigonometric functions sine and cosine, tangent and cotangent, and secant and cosecant of a right angle applies:

• Are not defined, and

Units

A right angle corresponds to the angle different dimensions:

From 5 July 1970 to 29 November 1973 onwards, in addition to the full angle (360 degrees) and the right angle with the unit symbol ∟ in Germany is a legal entity. Until 31 December 1996, the right angle was legal unit in Switzerland.

Construction

Tools for drawing rectangular lines, for example, in school, a mathematical paper or a protractor. To construct with ruler and compass see Lot ( mathematics). In technical drawing on the drawing board a character head is used with signs rails. In the metal-and woodworking craft a square or a doctrine is used to dimension right angle.

Various mechanical devices have been developed over time for the construction of right angle over longer distances. In the architecture of ancient Egypt and the Middle Ages were of this calculation ropes, for example, a twelve knot cord used. In the Roman civil engineering Groma a stakeout right angle was used in the Limitation of settlements, came more recently this an Oldham used. In geodesy comes in cadastral surveys with the Orthogonalverfahren a rectangular prism or a theodolite used. Today, these devices have been largely replaced by the electro- optical distance meter, such as, for example, total stations.

In practice, one always gets so of course only approximations to the geometric concept of the right angle.

Two hair angle

Engineer on the drawing board

A stop angle

Marking and Coding

To identify the right angle in a drawing both legs of the angle connecting quarter circle with a dot is used in the German-speaking as well as a number of other European countries. Occasionally, the item will also be omitted. In English-speaking one, both legs of the angle and connecting them with a small square (or at oblique parallelogram representation ) forming second right angle is shown for identification.

In the character set right angles are defined as follows and codes:

The characters ∟ for the right angle was first described by the Greek mathematician Pappus in the fourth century. AD used.