Förstå randvinkelsatsen: Skillnad mellan sidversioner
Hakan (diskussion | bidrag) |
Hakan (diskussion | bidrag) |
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| Rad 10: | Rad 10: | ||
Let ''O'' be the center of a circle. Choose two points on the circle, and call them ''V'' and ''A''. Draw line ''VO'' and extended past ''O'' so that it intersects the circle at point ''B'' which is [[diametrically opposite]] the point ''V''. Draw an angle whose vertex is point ''V'' and whose sides pass through points ''A'' and ''B''. | Let ''O'' be the center of a circle. Choose two points on the circle, and call them ''V'' and ''A''. Draw line ''VO'' and extended past ''O'' so that it intersects the circle at point ''B'' which is [[diametrically opposite]] the point ''V''. Draw an angle whose vertex is point ''V'' and whose sides pass through points ''A'' and ''B''. | ||
Angle ''BOA'' is a | Angle ''BOA'' is a central angle; call it ''θ''. Draw line ''OA''. Lines ''OV'' and ''OA'' are both [[radius|radii]] of the circle, so they have equal lengths. Therefore triangle ''VOA'' is isosceles, so angle ''BVA'' (the inscribed angle) and angle ''VAO'' are equal; let each of them be denoted as ''ψ''. | ||
Angles ''BOA'' and ''AOV'' are | Angles ''BOA'' and ''AOV'' are supplementary angle|supplementary. They add up to 180°, since line ''VB'' passing through ''O'' is a straight line. Therefore angle ''AOV'' measures 180° − θ. | ||
It is known that the three angles of a | It is known that the three angles of a triangle add up to 180°, and the three angles of triangle ''VOA'' are: | ||
: 180° − θ | : 180° − θ | ||
| Rad 22: | Rad 22: | ||
Therefore | Therefore | ||
:< | :ψ + 180<sup>o</sup> - θ = 180<sup>o</sup> | ||
Subtract 180° from both sides, | Subtract 180° from both sides, | ||
: | :2 ψ = θ | ||
where ''θ'' is the central angle subtending arc ''AB'' and ''ψ'' is the inscribed angle subtending arc ''AB''. | where ''θ'' is the central angle subtending arc ''AB'' and ''ψ'' is the inscribed angle subtending arc ''AB''. | ||
Versionen från 21 mars 2012 kl. 10.51
Komplementvinkeln till centrumvinkeln ger att de två lika vinklarna i den likbenta triangeln är lika stora (180-(180-x)). Det ger att randvinkeln = halva centrumvinkeln vilket ger randvinkelsatsen. VSB.
Bild från GeoGebra
Inscribed angles where one chord is a diameter
texten från Wikipedia.
Let O be the center of a circle. Choose two points on the circle, and call them V and A. Draw line VO and extended past O so that it intersects the circle at point B which is diametrically opposite the point V. Draw an angle whose vertex is point V and whose sides pass through points A and B.
Angle BOA is a central angle; call it θ. Draw line OA. Lines OV and OA are both radii of the circle, so they have equal lengths. Therefore triangle VOA is isosceles, so angle BVA (the inscribed angle) and angle VAO are equal; let each of them be denoted as ψ.
Angles BOA and AOV are supplementary angle|supplementary. They add up to 180°, since line VB passing through O is a straight line. Therefore angle AOV measures 180° − θ.
It is known that the three angles of a triangle add up to 180°, and the three angles of triangle VOA are:
- 180° − θ
- ψ
- ψ.
Therefore
- ψ + 180o - θ = 180o
Subtract 180° from both sides,
- 2 ψ = θ
where θ is the central angle subtending arc AB and ψ is the inscribed angle subtending arc AB.