A tangent segment from an external point to a circle has length 12. The circle's radius is 9. What is the distance from the external point…

A radius drawn to the point of tangency is perpendicular to the tangent segment. That forms a right triangle with legs $9$ (the radius) and $12$ (the tangent), and the distance from the external point to the center is the hypotenuse: $\sqrt{9^2+12^2}=\sqrt{81+144}=\sqrt{225}=15$, choice A. This is a $3$-$4$-$5$ triangle scaled by $3$. Choice B ($21$) adds the legs. Choice C ($\sqrt{63}$) treats $12$ as the hypotenuse, $\sqrt{144-81}$. Choice D ($3$) subtracts the two lengths. The distance to the center must exceed both $9$ and $12$.