Tendon (geometry)

Circle with tendon (red), arc (green) and peripheral angles phi (φ) and psi (ψ).

A chord of a flat curve is a line connecting two points on the curve. It is therefore the part of a secant that lies between the two curve points.

Tendon on circle

The chord of a circle divides the circle into two usually unequal arcs and , in each of which the set of peripheral angles applies: All triangles with the chord as the base and a third point on one of the arcs or have equal angles or . ${\displaystyle b_{1}}$ ${\displaystyle b_{2}}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle C}$ ${\displaystyle b_{1}}$ ${\displaystyle b_{2}}$ ${\displaystyle C}$ ${\displaystyle \phi }$ ${\displaystyle \psi }$

If the chord runs through the center of the circle , it is called the diameter . The peripheral angle is then a right angle ( Thales theorem ). ${\displaystyle M}$

The following applies to the chord length ${\displaystyle s}$

${\displaystyle s=2r\sin \left({\frac {\alpha }{2}}\right)}$

and because of as well ${\displaystyle 2\psi =\alpha \,}$ ${\displaystyle 2\varphi =360^{\circ }-\alpha \,}$

${\displaystyle s=2r\sin \varphi {\text{ und }}s=2r\sin \psi .}$

Historically, the chord length was no longer the usual trigonometric function Chord calculated. In the past, the perpendicular of the tendon to the center of the circle was called the apothema . The extension of the plumb bob over the chord to the edge of the circle was called sagitta . The lengths of apothema and sagitta together make up the circle radius.

Al-Battânîs (* between 850 and 869, † 929) was the first to use the sinus instead of geometric tendons.

See also

• String set
• Butterfly set
• Zindler curve

literature

• Schülerduden: Mathematik I, Dudenverlag, 8th edition, Mannheim 2008, pp. 415–418

Web links

Commons : Chord (geometry)  - collection of images, videos and audio files