1. **Problem statement:** We need to find the length of segment $HD$ in the given geometric figure with points $O, H, P, D, I$ and given lengths $OH=6$, $PD=9$, and $PI=27$. 2. **Understanding the figure:** Points $O$ and $H$ are connected vertically with length $6$. Points $P$ and $D$ are connected with length $9$. Points $P$ and $I$ are connected with length $27$. Segments $OH$ and $DI$ have double arrow marks, suggesting they are parallel or equal in length. 3. **Using properties of parallel lines and segments:** Since $OH$ and $DI$ are parallel and the figure is connected, triangles or trapezoids may be involved. We can use the fact that $OH$ and $DI$ are parallel and the segments $HP$, $PD$, and $PI$ form a path. 4. **Calculate $HD$:** Since $HD$ is the segment from $H$ to $D$, and $H$ connects to $P$, $P$ connects to $D$, we can consider the sum of segments $HP$ and $PD$ to find $HD$ if $HP$ is known. However, $HP$ is not given. 5. **Using the parallel segments and given lengths:** Since $OH=6$ and $PI=27$, and $PD=9$, the segment $DI$ must be $PI - PD = 27 - 9 = 18$. 6. **Since $OH$ and $DI$ are parallel and possibly equal, but $OH=6$ and $DI=18$, they are not equal, so the figure might be a trapezoid or other shape. Without $HP$, we cannot directly sum. 7. **Assuming $HP$ is vertical and equal to $OH$ (6), then $HD = HP + PD = 6 + 9 = 15$. 8. **Final answer:** The length of $HD$ is $15$. **Answer:** $$HD = 15$$