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The problem shows two similar triangles, and we are asked to find the value of \( y \) in terms of \( x \).

### Step-by-step Solution:

1. **Similarity of Triangles:**
   Since the triangles are similar, their corresponding sides are proportional. This means that the ratio of corresponding sides in one triangle equals the ratio of corresponding sides in the other triangle.

2. **Identify Corresponding Sides:**
   - The side of length \( 2 \) in the smaller triangle corresponds to the side of length \( x \) in the larger triangle.
   - The side of length \( 3 \) in the smaller triangle corresponds to the side of length \( 6 \) in the larger triangle.
   - The side of length \( x \) in the smaller triangle corresponds to the side of length \( y \) in the larger triangle.

3. **Set Up Proportions:**
   Using the ratio of the corresponding sides, we can set up a proportion:

   \[
   \frac{2}{x} = \frac{3}{6}
   \]

4. **Solve for \( x \):**
   Simplifying the ratio \( \frac{3}{6} \), we get:

   \[
   \frac{3}{6} = \frac{1}{2}
   \]

   Therefore, the equation becomes:

   \[
   \frac{2}{x} = \frac{1}{2}
   \]

   Cross-multiply to solve for \( x \):

   \[
   2x = 4 \quad \Rightarrow \quad x = 2
   \]

5. **Find \( y \) in terms of \( x \):**
   Now that we know \( x = 2 \), we can find \( y \). From the similarity, the ratio between the corresponding sides of \( x \) and \( y \) must match the ratio between 3 and 6. So we set up the proportion:

   \[
   \frac{x}{y} = \frac{3}{6} = \frac{1}{2}
   \]

   Solving for \( y \):

   \[
   x = \frac{y}{2}
   \]

6. **Conclusion:**
   Plugging \( x = 2 \) into the equation, we get:

   \[
   y = x + 3
   \]

Thus, the answer is \( \boxed{H} \).

Let me know if you'd like further clarifications or have any other questions! Here are some related questions you can explore:

1. How do you determine the similarity of triangles?
2. What is the significance of corresponding sides in similar triangles?
3. Can two triangles be similar if their angles differ but sides are proportional?
4. How does the ratio of the sides affect the area of similar triangles?
5. What is the relationship between angles and sides in similar triangles?

**Tip:** When working with similar triangles, always ensure that you correctly match the corresponding sides and angles before setting up proportions.

In the following figure, the two triangles are similar. The lengths of their sides are shown. What is the value of y in terms of x?

In this geometry problem, we are asked to find the value of y in terms of x by using the concept of similar triangles and the proportionality of their corresponding sides.

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