Difference between revisions of "Inscribed angle"
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In a given [[circle]], an '''inscribed angle''' is an [[angle]] whose [[vertex]] lies on the circle and each of whose [[side]]s either [[intersect]]s the circle in a second point (and so includes a [[chord]] of the circle) or is [[tangent line | tangent]] to the circle. This latter case is sometimes referred to as a [[tangent-chord angle]]. | In a given [[circle]], an '''inscribed angle''' is an [[angle]] whose [[vertex]] lies on the circle and each of whose [[side]]s either [[intersect]]s the circle in a second point (and so includes a [[chord]] of the circle) or is [[tangent line | tangent]] to the circle. This latter case is sometimes referred to as a [[tangent-chord angle]]. | ||
| − | { | + | <center><asy> |
| + | defaultpen(fontsize(8)); | ||
| + | int i, r=10;pair A=r*expi(-pi/2-pi/12), B=r*expi(-pi/2+pi/12), O=(0,0), P=B-5*expi(pi/12); | ||
| + | draw(Circle((0,0),r));draw(A--O--B--A);draw(B--P);for(i=-1;i<6;++i){draw(A--r*expi(pi*i/4)--B);} | ||
| + | dot(A^^B^^O^^P);label("A",A,(-1,-1));label("B",B,(1,-1));label("O",O,(0,1));label("P",P,(0,-1)); | ||
| + | </asy></center> | ||
The measure of an inscribed angle is equal to half of the measure of the [[arc]] it intercepts or subtends. Thus, in particular it does not depend on the location of the vertex on the circle. | The measure of an inscribed angle is equal to half of the measure of the [[arc]] it intercepts or subtends. Thus, in particular it does not depend on the location of the vertex on the circle. | ||
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[[Category:Geometry]] | [[Category:Geometry]] | ||
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Latest revision as of 21:47, 19 December 2007
In a given circle, an inscribed angle is an angle whose vertex lies on the circle and each of whose sides either intersects the circle in a second point (and so includes a chord of the circle) or is tangent to the circle. This latter case is sometimes referred to as a tangent-chord angle.
![[asy] defaultpen(fontsize(8)); int i, r=10;pair A=r*expi(-pi/2-pi/12), B=r*expi(-pi/2+pi/12), O=(0,0), P=B-5*expi(pi/12); draw(Circle((0,0),r));draw(A--O--B--A);draw(B--P);for(i=-1;i<6;++i){draw(A--r*expi(pi*i/4)--B);} dot(A^^B^^O^^P);label("A",A,(-1,-1));label("B",B,(1,-1));label("O",O,(0,1));label("P",P,(0,-1)); [/asy]](http://latex.artofproblemsolving.com/b/7/a/b7afb731d76b7598c668d86105ce3b2d9004d32b.png)
The measure of an inscribed angle is equal to half of the measure of the arc it intercepts or subtends. Thus, in particular it does not depend on the location of the vertex on the circle.
See also
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