To solve for $x$ in the triangle, we should analyze the similar triangles formed by the line segments parallel to the sides of the triangle.

Given:

• The largest triangle has a side of length $4$.
• The second triangle, smaller and within the largest triangle, has a corresponding side of length $3$.
• The smallest triangle, within the second triangle, has a corresponding side of length $1.5$.
• We are asked to find $x$, which is the corresponding side of the smallest triangle formed by the parallel lines.

Solution:

1. Identify the Similar Triangles:

   • The triangles are similar because their corresponding sides are parallel.

2. Set up the Proportion:

   • The ratio of the corresponding sides of similar triangles is equal.
   • For the two similar triangles: $\frac{x}{1.5} = \frac{3}{4}$

3. Solve for $x$:

   • Cross-multiply to solve for $x$: $x \cdot 4 = 3 \cdot 1.5$ $4x = 4.5$ $x = \frac{4.5}{4}$ $x = 1.125$

Therefore, $x = 1.125$.

Would you like further details on this solution, or do you have any other questions?

Here are some related questions that might interest you:

1. What are similar triangles, and how do they work?
2. How can proportions be used to solve for unknowns in geometry?
3. What other properties of triangles are useful in solving geometric problems?
4. Can this method be applied to other geometric shapes?
5. How does understanding ratios and proportions help in real-world applications?

Tip: When working with similar figures, always look for proportional relationships between corresponding sides to solve for unknown values.