14. (15 points) As shown in the figure, in a square $P Q R S$ with a side length greater than $20 \mathrm{~cm}$, there is a largest circle $O$. If a point $T$ on the circumference of the circle is $8 \mathrm{~cm}$ away from $P S$ and $9 \mathrm{~cm}$ away from $P Q$, what is the radius of circle $O$ in centimeters?
【Analysis】According to the problem, draw the figure as follows:
Let the radius of the circle be $x(x>10)$, then $T M=x-8, O M=T N=x-9$. In the right triangle $T M O$, according to the Pythagorean theorem, we have: $(x-8)^{2}+(x-9)^{2}=x^{2}$. Solve the equation.

【Solution】Solution: As shown in the figure,
Let the radius of the circle be $x(x>10)$, then $T M=x-8, O M=T N=x-9$,
In the right triangle $T M O$, according to the Pythagorean theorem, we have:
$$
(x-8)^{2}+(x-9)^{2}=x^{2}
$$

Simplifying, we get: $x^{2}-34 x+145=0$
That is $(x-5)(x-29)=0$
Thus, $x=5$ (discard) or $x=29$
Answer: The radius of circle $O$ is 29 cm.