1. **Problem statement:** Quadrilateral EFGH is a kite, and we need to find the length of segment $DH$. 2. **Properties of a kite:** A kite has two pairs of adjacent sides that are equal. Also, the diagonals intersect at right angles, and one diagonal bisects the other. 3. **Given information:** From the diagram, $EF = 8$, $FG = 17$, and $E$, $F$, $G$, $H$, $D$ are points with $D$ on diagonal $EG$. 4. Since $EFGH$ is a kite, $EF = EH = 8$ and $FG = GH = 17$. 5. The diagonal $EG$ is split by point $D$ such that $ED = DH$ because the diagonal $EG$ is bisected by the other diagonal $FH$ at $D$. 6. To find $DH$, we use the Pythagorean theorem on triangle $FDH$ where $FD$ is perpendicular to $EG$. 7. Since $EF = 8$ and $FG = 17$, and $D$ is the midpoint of $EG$, the length $DH$ equals $8$. **Final answer:** $$DH = 8$$