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Math·Geometry and Trigonometry·Circles

hard

A circle has center $O$. From a point $P$ outside the circle, a tangent segment $PT$ and a secant through points $A$ and $B$ are drawn, with $A$ between $P$ and $B$. If $PA=8$ and $AB=10$, what is the length of $PT$?

A

$4\sqrt{5}$

B

12

C

$6\sqrt{2}$

D

$12\sqrt{2}$