To solve for the length of $\overline{AC}$ in the kite $ABCD$, we will use the properties of the kite and the Pythagorean theorem. Given:

1. $AC \perp DB$ (diagonals are perpendicular)
2. $DE = EB = 16$ cm (diagonals bisect each other)
3. $DC = 9$ cm and $DA = 10$ cm

Since $AC$ and $DB$ are perpendicular, they form right triangles within the kite. We need to find the length of $\overline{AC}$, which is the length from $C$ to $A$. Since $DE = EB = 16$ cm, the total length of $DB$ is $32$ cm.

Now, let's calculate $\overline{AC}$:

1. In triangle $\triangle DEC$, we know: $DE = 16 \text{ cm}, \quad DC = 9 \text{ cm}$ We can use the Pythagorean theorem: $EC^2 + DE^2 = DC^2$ Substituting the known values: $EC^2 + 16^2 = 9^2$ $EC^2 + 256 = 81$ $EC^2 = 81 - 256 = -175$ There seems to be an issue. Let me recheck