In this image, we are dealing with two right triangles that are similar due to the shared angle at the top and the fact that both are right triangles. Given that these triangles are similar, their corresponding sides are proportional.

Here is the setup:

• In the smaller triangle, the legs are 3 and 4, and we need to find the hypotenuse, which is denoted as $x$.
• In the larger triangle, the hypotenuse is 6, and we know that the corresponding side of the larger triangle to the side of length 3 is 6.

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

$\frac{x}{6} = \frac{4}{3}$

Now solve for $x$:

$x = 6 \times \frac{4}{3}$ $x = 8$

Thus, the length of $x$ is 8.

Would you like any further clarification? Here are some additional questions to deepen your understanding:

1. How can we confirm that the triangles are indeed similar?
2. What is the relationship between the ratios of corresponding sides in similar triangles?
3. Can you use the Pythagorean theorem to check the side lengths of these triangles?
4. What would happen if the ratio of sides were reversed (e.g., a different triangle orientation)?
5. How would you handle a situation where the triangles are not right triangles but still similar?

Tip: Always look for corresponding sides when working with similar triangles, as their ratios will give you a way to solve for unknowns.