CRITICAL THINKING In the figure, $A B=12, B C=8$, $D E=6, P D=4$, and $A$ is a point of tangency. Find the radius of $\odot P$.

Step 1 ：Let the radius of the circle be \(r\). Since A is a point of tangency, AP is perpendicular to the tangent line AB. Therefore, triangle APB is a right triangle. We can use the Pythagorean theorem to find the length of AP: \(AP = \sqrt{AB^2 - r^2} = \sqrt{12^2 - r^2}\).

Step 2 ：Observe that triangle APD is similar to triangle CPB. We can use the similarity ratio to find the length of CP: \(\frac{CP}{AP} = \frac{CB}{PD} \Rightarrow CP = \frac{8r}{12 - r}\).

Step 3 ：Finally, we can use the Pythagorean theorem again to find the length of the radius \(r\) in triangle CPD: \(r^2 + CP^2 = CD^2 \Rightarrow r^2 + \left(\frac{8r}{12 - r}\right)^2 = 6^2\).

Step 4 ：Solving the equation for \(r\), we get \(r \approx 4\).

Step 5 ：\(\boxed{4}\) is the radius of the circle.