Geometry

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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?

  • Geometry -

    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°

  • Geometry -

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

  • Geometry -

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

  • Geometry -

    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.

  • Geometry -

    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.

  • Geometry -

    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.

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