To solve for $x$ in this image, we observe that this problem involves similar triangles.

Step 1: Identify the triangles

1. The larger triangle has sides $25$, $15$, and $3$.
2. The smaller triangle (upper portion) has sides $x-2$ (hypotenuse) and two sides proportional to the smaller segments.

By the property of similar triangles, the ratios of corresponding sides are equal.

Step 2: Set up the proportions

We use the ratio of the smaller triangle's base to the larger triangle's base, and its height to the larger triangle's height: $\frac{\text{Smaller Triangle Base}}{\text{Larger Triangle Base}} = \frac{\text{Smaller Triangle Height}}{\text{Larger Triangle Height}}$

The proportions are: $\frac{x-2}{25} = \frac{3}{15}$

Step 3: Simplify the ratio

The height ratio simplifies to: $\frac{3}{15} = \frac{1}{5}$

Thus, the equation becomes: $\frac{x-2}{25} = \frac{1}{5}$

Step 4: Solve for $x$

Cross-multiply to find $x$: $5(x-2) = 25$ $x-2 = 5$ $x = 7$

Final Answer:

$x = 7$

Would you like further explanation or visual clarification?

---

Here are some related questions to deepen understanding:

1. How do properties of similar triangles apply to real-world problems?
2. What are the criteria to identify similar triangles?
3. How can you verify proportionality in triangle geometry?
4. What happens if one side of the triangle changes—how does it affect the ratios?
5. How would this problem change if additional dimensions were provided?

Tip: Always double-check if the triangles are indeed similar by comparing their angles or proportional sides before solving.