The problem involves two similar triangles, and we are asked to find the value of $x$. Since the triangles are similar, the corresponding sides are proportional.

We are given the following side lengths:

• Triangle 1: $9$ (vertical side), $7.7$ (hypotenuse), and $5.9$ (base).
• Triangle 2: $6$ (vertical side), and $x$ (hypotenuse).

Using the property of similar triangles, we can set up the following proportion:

$\frac{7.7}{9} = \frac{x}{6}$

Now we solve for $x$:

$x = \frac{7.7}{9} \times 6$

First, calculate the ratio:

$\frac{7.7}{9} = 0.8556$

Now, multiply by 6:

$x = 0.8556 \times 6 = 5.1336$

Rounding to 1 decimal place:

$x \approx 5.1$

Thus, the value of $x$ is approximately $5.1$.

Do you need more details or have any questions?

Here are 5 related questions for further practice:

1. How do we know that the triangles in the diagram are similar?
2. What are other ways to solve for the unknown side of a right triangle?
3. How can you verify if two triangles are similar?
4. What are the key properties of similar triangles?
5. How do you apply the Pythagorean theorem to right triangles?

Tip: When dealing with similar triangles, always check if corresponding angles are equal and if the ratios of corresponding sides are proportional.