The image shows a geometric figure consisting of several triangles, and the question asks to find the value of $x$. The triangles $\triangle BEC$, $\triangle CED$, and $\triangle DAB$ are mentioned as being relevant to solving for $x$.

Given information:

• $AB = 8 \, \text{m}$
• $BC = 14 \, \text{m}$
• $AD = 15 \, \text{m}$
• $CE = 6 \, \text{m}$
• The length $x$ (side $CD$) is unknown.

Plan:

Since three triangles are indicated and likely related, we might need to apply properties of similarity, the Pythagorean theorem, or a trigonometric relationship between the triangles to solve for $x$. These triangles could be sharing common angles or sides, which will help set up the necessary equations.

I'll calculate and analyze this further.The value of $x$, representing the length of side $CD$, is approximately:

$x = -\sqrt{413 - 136\sqrt{10}} \, \text{m}$

Since a negative length doesn't make sense in this context, it suggests that either a geometric constraint has been overlooked, or that the setup of the problem could be revisited for simplification.

Would you like further clarifications on the method or a detailed breakdown of the steps? Here are a few related questions to expand upon:

1. What is the significance of applying the Pythagorean theorem in non-right triangles?
2. How do triangle similarity properties help in solving for unknown sides?
3. Can you explain how to interpret negative solutions in geometric problems?
4. What other geometric theorems could be applicable to quadrilaterals?
5. Could the use of trigonometry be another approach to find $x$?

Tip: Always check the feasibility of your results when dealing with geometric shapes, especially when negative values appear in practical problems.