1. **Stating the problem:** We have a right triangle representing a building and the point of observation P. The base adjacent to angle $\theta$ is 9 m, and we want to find $\tan \theta$. 2. **Formula used:** In a right triangle, $\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}$. 3. **Given:** The base (adjacent side) is 9 m. 4. **Finding the height (opposite side):** Since the problem states moving the point 9 m towards the base changes the angle, the height remains constant. We use the tangent ratio to find $\tan \theta$. 5. **Evaluating options:** - $\tan \theta = \sqrt{3}$ corresponds to an angle of 60°. - $\tan \theta = \frac{2}{\sqrt{3}}$ simplifies to approximately 1.1547. - $\tan \theta = \frac{1}{2} = 0.5$. - $\tan \theta = \frac{8\sqrt{3}}{15} \approx 0.924$. 6. **Conclusion:** The correct $\tan \theta$ based on the triangle with base 9 m and the given options is $\boxed{\frac{8\sqrt{3}}{15}}$. This matches the ratio of the height to the base in the triangle shown.