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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 $\angle TPO = 33^{\circ}$ and $PT = 10$, then what is the radius of the circle?

 Jan 8, 2024
 #1
avatar+128448 
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                                         T    

                                                      10

                   O                                      33   P

                                                      10

                                         U 

 

Connect  OT , TP  and OP

 

Triangle  OTP will be a right triangle with angle OTP  = 90°  and OT  the radius of the circle and angle POT = 90 - 33  =  57°

 

By the Law of Sines

 

PT / sin 57° = OT / sin 33°

 

10 sin 33° / sin 57°  = OT ≈  6.5  = circle radius

 

 

cool cool cool

 Jan 8, 2024

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