Geometry
posted by Amy .
Points $A$ and $B$ are on a circle centered at $O$, and point $P$ is outside the circle such that $\overline{PA}$ and $\overline{PB}$ are tangent to the circle. If $\angle OPA = 32^{\circ}$, then what is the measure of minor arc $AB$, in degrees?

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According to the properties of tangents to circles,
AO is perpendicular to PA making angle PAO = 90°,
thus angle angle POA = 9032 = 58°
The same property is true for angle POB
since PA = PB , angle POB = 58°
Then angle AOB = 116° and the arc AB is subtended by a central angle of 116°
or
PAOB is a quadrilateral with 2 right angles
thus angle AOB = 360°  2(90)°  64° = 116° 
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