Thank you for the correction. Let's recalculate the value of $x$ given that $DE = 2\sqrt{5}$.

1. The lengths are:

   • $AB = 6$
   • $BC = 4$
   • $CD = 3$
   • $DE = 2\sqrt{5}$
   • $x$ is the length of segment $AE$

2. Calculate $AD$ using the Pythagorean theorem in $\triangle ABD$: $AD = \sqrt{AB^2 + BD^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}$

3. Now calculate $AE$ in $\triangle ADE$: $AE = \sqrt{AD^2 + DE^2} = \sqrt{(3\sqrt{5})^2 + (2\sqrt{5})^2} = \sqrt{45 + 20} = \sqrt{65}$

Therefore, the length of $x$ is: $x = \sqrt{65} \approx 8.06$

So, $x \approx 8.06$.

Would you like further details or have any questions?

Here are 8 related questions:

1. What are the properties of a trapezoid?
2. How do you apply the Pythagorean theorem in a trapezoid?
3. What are the characteristics of similar triangles?
4. How do you find the area of a trapezoid?
5. What is the significance of right angles in geometric shapes?
6. How do you determine the length of a segment in a polygon?
7. What other methods can be used to solve for unknown sides in a trapezoid?
8. How does the concept of proportion apply to trapezoids?

Tip: Always double-check the values provided in geometry problems to ensure accuracy in calculations.