To calculate the length of $AC$, we use the Pythagorean theorem. Here’s the step-by-step explanation:

Given:

1. $AC$ is perpendicular to $DB$.
2. $D$ is 8 cm away from $B$, so the total $DB = 8 + 8 = 16 \, \text{cm}$.
3. $DE = EB = 5 \, \text{cm}$, and $AC$ bisects $DB$ at $E$.

Solution:

1. Determine $AE$ and $EC$: Since $AC$ bisects $DB$, the lengths are split equally. Hence, $AE = EC$.

2. Apply the Pythagorean theorem to triangle $AED$: $AD^2 = AE^2 + DE^2$ Substitute $AD = 8 \, \text{cm}$ and $DE = 5 \, \text{cm}$: $8^2 = AE^2 + 5^2$ $64 = AE^2 + 25$ $AE^2 = 39$ $AE = \sqrt{39} \approx 6.2 \, \text{cm}.$

3. Since $AE = EC$, the total length of $AC$ is: $AC = 2 \times AE = 2 \times 6.2 = 12.4 \, \text{cm}.$

Final Answer:

$AC \approx 12.4 \, \text{cm}.$

Let me know if you need further clarification!

Here are 5 related questions:

1. How is the Pythagorean theorem applied to triangles in kites?
2. Why is $AC$ split equally in the context of this problem?
3. What are the properties of a kite that help solve this problem?
4. How would the calculation change if $DE \neq EB$?
5. Can this problem be generalized to other quadrilaterals?

Tip: Always double-check which line segments are perpendicular or bisected when working with kites!