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?

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15 answers
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Answer 1

Reiny answered
8 years ago

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 2

AoPS answered
7 years ago

Stop cheating! Please use our own message boards if you get stuck.

Answer 3

a nony mouse answered
7 years ago

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

Answer 4

AoPS User answered
7 years ago

AoPS message boards are used for the same use as this website is used for. AoPS can track your IP address, and find out who you are. Next time, use the message boards.

Answer 5

AoPS answered
6 years ago

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 6

F AOPS answered
6 years ago

all u are fake

Answer 7

to them aops haters answered
5 years ago

DO we care

Answer 8

AoPS answered
4 years ago

i am real aops hahaha we actually reccomend using this cause aops message boards are very unrealiable so eys use this we encourage

Answer 9

AoPS answered
4 years ago

Please stop cheating!

Answer 10

AoPS answered
4 years ago

lol guys...thats not the right answer anyway

Answer 11

AoPS Person answered
4 years ago

Right that is wrong

Answer 12

Who cares? answered
4 years ago

Bruh do AoPS have trackers installed in everyone's devices? lol just asking.

Answer 13

bruh yall dum answered
3 years ago

yall know that aint the real aops right

Answer 14

I'm so sorry answered
3 years ago

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

Answer 15

AoPS answered
1 year ago

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?

Just use ordinary keyboard keys

Stop cheating! Please use our own message boards if you get stuck.

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

AoPS message boards are used for the same use as this website is used for. AoPS can track your IP address, and find out who you are. Next time, use the message boards.

all u are fake

DO we care

i am real aops hahaha we actually reccomend using this cause aops message boards are very unrealiable so eys use this we encourage

Please stop cheating!

lol guys...thats not the right answer anyway

Right that is wrong

Bruh do AoPS have trackers installed in everyone's devices? lol just asking.

yall know that aint the real aops right

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

may your nuts forever be shriveled