In the image, we see two triangles: $\triangle ABC$ and $\triangle DBE$, which are stated to be similar ($\triangle ABC \sim \triangle DBE$).

Step-by-step solution:

1. Similarity of triangles: Since $\triangle ABC \sim \triangle DBE$, corresponding sides of the two triangles are proportional. The sides of $\triangle ABC$ are $AB$, $BC$, and $AC$. The sides of $\triangle DBE$ are $DB$, $BE$, and $DE$.

2. Labeling the corresponding sides:

   • $AB$ (side in $\triangle ABC$) corresponds to $DB$ (side in $\triangle DBE$),
   • $BC$ corresponds to $BE$,
   • $AC$ corresponds to $DE$.

3. Setting up the proportion using the lengths of the sides given:

   • $\frac{AB}{DB} = \frac{BC}{BE} = \frac{AC}{DE}$.

   Using the numbers from the image:

   • $AB = 10$, $DB = 6x$,
   • $BC = 12$, $BE = 8x - 4$,
   • $AC = 6$, $DE = unknown$.

4. Using the proportion for $AB$ and $DB$: $\frac{AB}{DB} = \frac{10}{6x}.$

   Using the proportion for $BC$ and $BE$: $\frac{BC}{BE} = \frac{12}{8x - 4}.$

5. Solving for $x$ by setting up an equation: $\frac{10}{6x} = \frac{12}{8x - 4}.$

   Cross-multiply to get: $10(8x - 4) = 12(6x),$ $80x - 40 = 72x.$

   Simplify and solve for $x$: $80x - 72x = 40,$ $8x = 40,$ $x = 5.$

Thus, the value of $x$ is 5.

Would you like more details or have any further questions?

Here are 5 related questions you can explore:

1. How can you verify that two triangles are similar?
2. What is the significance of corresponding sides in similar triangles?
3. How do you apply proportions to solve for unknown sides in similar triangles?
4. What is the Angle-Angle (AA) criterion for triangle similarity?
5. Can similarity of triangles be used in real-life applications like surveying?

Tip: Always ensure the corresponding sides of two similar triangles are identified correctly before setting up the proportion equation.