This problem involves solving for the variable $x$ using similar triangles.

In the image, we have two similar right triangles. Since the triangles are similar, the sides of the smaller triangle will have the same ratios as the sides of the larger triangle. The sides given are:

• The smaller triangle has a leg of length $8$ and the leg along the height $x$.
• The larger triangle has a base of $32$, which corresponds to the total base of the large triangle.

Set up the proportion:

Since the triangles are similar, we can set up the following proportion using corresponding sides:

$\frac{x}{8} = \frac{8}{32}$

Solve for $x$:

$\frac{x}{8} = \frac{1}{4}$

Now, multiply both sides by 8 to solve for $x$:

$x = \frac{1}{4} \times 8 = 2$

Thus, the value of $x$ is $\boxed{2}$.

Would you like more details, or have any other questions?

Here are 5 questions to deepen your understanding:

1. What properties define similar triangles?
2. Can you always apply proportions to similar triangles regardless of their size?
3. How would the equation change if the height of the larger triangle was given?
4. What is the significance of the 90-degree angle in this context?
5. How does the Pythagorean Theorem apply to right triangles like these?

Tip: When working with similar figures, always remember that their corresponding sides are proportional.