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?

Question

Generate an image of a geometric scenario. There is a circle at the center of which is point O. On the edge of the circle, there are two points A and B. Outside of the circle, there is another point P. The lines PA and PB are tangent to the circle forming an angle OPA of 32 degrees. The image is meant to illustrate the minor arc between points A and B on the circle, while maintaining the angle OPA at 32 degrees.

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?

Answer 1

AoPS Admin and AoPS User, we appreciate that you are trying to stop people from cheating on AoPS homework, but unfortunately, no, we cannot track your IP address, nor do we care to do so. It is your choice if you decide to waste the homework questions, which are specifically designed to help you. And yes, if you're stuck the message boards are always open or you can privately ask your teacher if it is rather personal.

Answer 2

lol i feel like the AoPS account is fake and looking for answers too

Answer 3

all u are fake

Answer 4

lol guys...thats not the right answer anyway

Answer 5

Just use ordinary keyboard keys

According to the properties of tangents to circles,
AO is perpendicular to PA making angle PAO = 90°,
thus angle angle POA = 90-32 = 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°

Answer 6

Right that is wrong

Answer 7

DO we care

Answer 8

may your nuts forever be shriveled

Answer 9

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Answer 10

I'm sorry AoPS if this is helping them cheat, but this serves the same purpose as the massage board and there is a high chance that nobody in the message board helped them. The correct answer is 128 degrees