To find the length $x$ in the given right triangle, we can use the properties of similar triangles. The two triangles in the image are similar right triangles. This means the ratios of the corresponding sides are equal.

Given the lengths in the two triangles:

• For the larger triangle: the legs are $5$ and $x$.
• For the smaller triangle: the legs are $2.5$ and $2$.

Since the triangles are similar, the ratio of the corresponding sides should be equal: $\frac{x}{2.5} = \frac{5}{2}$

We can solve this proportion to find $x$: $\frac{x}{2.5} = \frac{5}{2}$

Cross-multiplying to solve for $x$: $x \cdot 2 = 5 \cdot 2.5$

$2x = 12.5$

$x = \frac{12.5}{2}$

$x = 6.25$

So, the length $x$ is $6.25$.

Would you like more details or have any questions about this solution?

Here are 5 related questions that expand on this information:

1. What are the properties of similar triangles?
2. How can you determine if two triangles are similar?
3. Can you solve for the length of the hypotenuse in each triangle?
4. How would you use the Pythagorean theorem to verify the solution?
5. What other methods can be used to find the lengths of sides in similar triangles?

Tip: When solving problems involving similar triangles, always remember to set up the correct ratio of corresponding sides before solving for the unknown lengths.