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urp:circles [2021-10-13]
nerf_herder created
urp:circles [2021-11-19] (current)
nerf_herder
Line 1: Line 1:
 ====Circles==== ====Circles====
 +{{ :​urp:​circles.jpg?​400|}}
   * __chord__ has two endpoints on a circle   * __chord__ has two endpoints on a circle
   * __secant__ is a line that contains a chord, but extends beyond the circle   * __secant__ is a line that contains a chord, but extends beyond the circle
 +  * __tangent__ is a line that touches a circle at one point (a secant with chord length of zero)
   * __Intercepted arc__ is the part of the circle contained within the two lines   * __Intercepted arc__ is the part of the circle contained within the two lines
-  * __Central angle__ is angle of two lines from the center of the circle +  * __Central angle__ is angle of two lines from the center of the circle, it has the same degrees as the intercepted arc 
-  * __Inscribed/​Interior angle__ has two points and the vertex all on the circle itself +  * __Inscribed/​Interior angle__ has two points and the vertex all on the circle itself, it has 1/2 the degrees of the intercepted arc
-  * __central angle__ is same as the degrees of intercepted arc+
  
-if two chords ABCD intersect at P, then AP * PB = CP * PD+In the image, P is center point of the circleline C is a chord and line S is a secant.
  
-Interior angle = 1/2 of intercepted arc +**intersecting ​chord theorem** - If two chords AB, CD intersect at P (not necessarily ​the center), then AP * PB CP * PD
-  all inscribed angles going to the two same points on the circle have the same angle +
-  angle of intersecting ​secants ​theorem:  +
-      angle formed by the secant intersection ​(opposite arc - adjacent arc)/2+
  
-An angle outside ​the circle ​with two secants ​(or tangentswill have an angle +All inscribed angles going to the two same points on the circle (but from different vertices) have the same angle
-  that is 1/2 * (difference of the intercepted arcs)+
  
-coterminal angle - the rest of the circle outside the angle.  +The **intersecting secants theorem**, for secants or tangents that intersect outside of the circle says the angle formed by the secant intersection = (opposite arc - adjacent arc)/2 
-  eg. angle of 30', the coterminal is 330'+ 
 +**coterminal angle** - the rest of the circle outside the angle.  
 +  eg. angle of 30°, the coterminal is 330°
   in radians, use the absolute value.  ​   in radians, use the absolute value.  ​
-      ​2pi - |angle|, or |angle| - 2pi for the negative angle +      ​2π - |angle|, or |angle| - 2π for the negative angle
  
-area of an arc:+**area of an arc** (from a central angle):
   a = rad*r²/​2 ​ .... comes from a = πr² for full circle, and the    a = rad*r²/​2 ​ .... comes from a = πr² for full circle, and the 
                       proportion of a circle in the arc is rad/2π                       proportion of a circle in the arc is rad/2π
                       so a = (rad/​2π)*(πr²)                       so a = (rad/​2π)*(πr²)
  
 +**length of arc**, given length of chord and radius:
 +  d = 2*r*sin(a/​2r) ​
 +     a = length of arc, d = length of chord, r = radius
 +     
 +**circumcircle** - a circle that touches the vertices of an polygon
 +   ​circumcircle of a triangle - has the center at the point where the
 +      perpendicular bisectors of the triangle'​s sides meet. 
 +      (might be outside the triangle).
 +   To find the radius: https://​mathworld.wolfram.com/​Circumradius.html
 +   
 +-------
 Back to [[geometry]] or [[math]] page. Back to [[geometry]] or [[math]] page.
urp/circles.1634102799.txt.gz · Last modified: 2021-10-13 by nerf_herder