The image shows two trapezoids, and the problem likely involves using the fact that these trapezoids are similar. In similar figures, corresponding sides are proportional.

The side lengths of the trapezoids are:

• Left trapezoid: $x+6$, 6, and 12
• Right trapezoid: 9, 21, and 18

Since these trapezoids are similar, we can set up the proportion between corresponding sides. We'll pair the sides of the trapezoids:

1. $\frac{6}{9}$ (corresponding smaller sides)
2. $\frac{12}{18}$ (corresponding larger sides)
3. $\frac{x+6}{21}$ (corresponding sides with $x+6$)

Now, let's solve for $x$ using these proportions.

$\frac{x+6}{21} = \frac{6}{9}$

Simplify $\frac{6}{9}$:

$\frac{6}{9} = \frac{2}{3}$

Now set up the equation:

$\frac{x+6}{21} = \frac{2}{3}$

Cross-multiply:

$3(x+6) = 2(21)$

Simplify:

$3(x+6) = 42$

$x+6 = 14$

$x = 14 - 6$

$x = 8$

The value of $x$ is 8.

Would you like more details on this process or have any other questions?

Here are five related questions to expand your understanding:

1. How do you determine when two shapes are similar?
2. What are the properties of trapezoids that make them similar?
3. Can you solve for the height of the trapezoid if given additional information?
4. How would the solution change if the sides were not proportional?
5. Can you derive a general formula for $x$ in similar trapezoids?

Tip: When working with similar figures, always ensure the corresponding sides are aligned properly before setting up proportions!