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# AREA CIRC HARD

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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.

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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°

CPhill  Aug 26, 2017

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