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

#1**+1 **

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