To calculate the length of $AC$, follow these steps:

1. Identify the structure of the kite:

   • $AC$ and $DB$ are perpendicular (intersecting at $E$).
   • $DE = EB = 8 \, \text{cm}$.
   • $DC = 9 \, \text{cm}$ and $DA = 10 \, \text{cm}$.

2. Split the kite into two right triangles:

   • Triangle $\triangle ADE$:
     • $DE = 8 \, \text{cm}$, $DA = 10 \, \text{cm}$.
   • Triangle $\triangle CBE$:
     • $EB = 8 \, \text{cm}$, $CB = 9 \, \text{cm}$.

3. Find the lengths $AE$ and $CE$ using the Pythagorean Theorem:

   • In $\triangle ADE$: $AE = \sqrt{DA^2 - DE^2} = \sqrt{10^2 - 8^2} = \sqrt{100 - 64} = \sqrt{36} = 6 \, \text{cm}.$
   • In $\triangle CBE$: $CE = \sqrt{CB^2 - EB^2} = \sqrt{9^2 - 8^2} = \sqrt{81 - 64} = \sqrt{17} \approx 4.1 \, \text{cm}.$

4. Calculate the total length of $AC$: $AC = AE + CE = 6 + 4.1 = 10.1 \, \text{cm}.$

Final Answer:

$AC \approx 10.1 \, \text{cm}.$

Do you want me to explain any step in more detail or have other questions?

Relative Questions for Further Exploration:

1. How do you apply the Pythagorean Theorem in 3D shapes or objects?
2. What are the properties of a kite that differentiate it from other quadrilaterals?
3. How can you calculate the area of a kite using diagonals?
4. What are some real-life applications of solving right triangles in geometry?
5. How would the calculation change if one side length of the kite was unknown?

Tip: Always label your triangle sides clearly when solving problems involving the Pythagorean Theorem to avoid confusion.