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Points $A$ and $B$ are on circle $O$ such that arc $AB$ is 80 degrees. A circle is constructed that passes through $A$, $B$, and $O$. Find the measure of arc $AOB$ on this circle.

AdminMod2  Aug 26, 2017
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Draw OA , OB and AB

 

Now since minor arc AB  = 80°....then, in triangle AOB,   since angle AOB is a central angle of the larger  circle  subtending minor arc AB, it  has the same measure.

 

And OA  = OB  since they are both radii of the larger circle....but if OA  = OB....then the angles opposite these sides in triangle AOB - namely angles OAB and OBA - are equal.... 

 

Thus....  OAB  = OBA  =   [ 180 - 80 ] / 2   = 50°

 

But angles OAB and OBA are inscribed angles in the smaller circle

And OAB  intercepts minor arc OB.....and its measure is twice that of OAB  =100°

 

Likewise....OBA  intercepts minor arc OA....and its measure is twice that of OBA  = 100°

 

Thus  minor arc OA  + minor arc OB  =  arc AOB

And    100° + 100°  =  arc AOB  = 200°

 

 

cool cool cool

CPhill  Aug 26, 2017

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