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Points $T$ and $U$ lie on a circle centered at $O$, and point $P$ is outside the circle such that $\overline{PT}$ and $\overline{PU}$ are tangent to the circle.  If $\angle TOP = 45^{\circ}$, then what is the measure of minor arc $TU$, in degrees?

 Jun 6, 2024
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PT = PU  )tangents form a point outside a circle are equal_

PO = PO

TO = UO  =   radii of the same  circle

So, by SSS  triangle TOP   is congruent to triangle UOP

So angleTOP  = angle UOP  = 45

So angle TOU = 90

This is a  central angle intercepting minor arc TU  ...so minor arc TU  = 90

 

cool cool cool

 Jun 6, 2024

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