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urp:circles
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urp:circles [2021-10-16] nerf_herder |
urp:circles [2021-11-19] nerf_herder |
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| ====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 AB, CD intersect at P, then AP * PB = CP * PD | + | In the image, P is center point of the circle, line 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 tangents) will have an angle that is 1/2 * (difference of the intercepted arcs) | + | All inscribed angles going to the two same points on the circle (but from different vertices) have the same 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 | ||
| **coterminal angle** - the rest of the circle outside the angle. | **coterminal angle** - the rest of the circle outside the angle. | ||
| Line 22: | Line 21: | ||
| 2π - |angle|, or |angle| - 2π for the negative angle | 2π - |angle|, or |angle| - 2π for the negative angle | ||
| - | + | **area of an arc** (from a central angle): | |
| - | **area of an arc**: | + | |
| 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). | ||
| + | |||
| + | ------- | ||
| Back to [[geometry]] or [[math]] page. | Back to [[geometry]] or [[math]] page. | ||
urp/circles.txt · Last modified: 2021-11-19 by nerf_herder
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