https://file.aiquickdraw.com/mathbot/file/1727702223527v7eglqok.jpg

To solve for \( x \), we need to analyze the two triangles given in the image.

### Step 1: Similarity of Triangles
From the image, it appears that the two triangles \( \triangle HNC \) and \( \triangle BMD \) are similar because their angles are congruent, which we can assume based on their positions and structure.

For similar triangles, corresponding sides are proportional. This gives us the proportion:
\[
\frac{HN}{BM} = \frac{NC}{MD} = \frac{HC}{BD}
\]

### Step 2: Corresponding Sides
From the image, we have the following lengths:
- \( HN = 40 \)
- \( NC = 3x - 3 \)
- \( HC \) (hypotenuse) corresponds to the side \( BD \), and is labeled as \( x + 7 \).

Similarly, for the smaller triangle \( \triangle BMD \):
- \( BM = 24 \)
- \( MD = x + 7 \)

### Step 3: Set Up Proportion
Using the similarity ratio between the triangles:

\[
\frac{HN}{BM} = \frac{NC}{MD}
\]

Substitute the given values:
\[
\frac{40}{24} = \frac{3x - 3}{x + 7}
\]

### Step 4: Solve the Proportion
Now, cross-multiply to solve for \( x \):
\[
40(x + 7) = 24(3x - 3)
\]

Expanding both sides:
\[
40x + 280 = 72x - 72
\]

### Step 5: Isolate \( x \)
Bring all terms involving \( x \) to one side and constants to the other side:
\[
280 + 72 = 72x - 40x
\]
\[
352 = 32x
\]

Solve for \( x \):
\[
x = \frac{352}{32} = 11
\]

### Final Answer:
The value of \( x \) is \( 11 \).

Would you like further details or have any questions?

### 5 Related Questions:
1. How do you verify if two triangles are similar?
2. What are the properties of similar triangles?
3. Can you explain the concept of proportional sides in similar triangles?
4. How would the solution change if the triangles were not similar?
5. What are other methods for solving triangle-related problems?

### Tip:
Always check if triangles are similar by verifying the congruence of their angles or the proportionality of their corresponding sides.

Learn how to solve for x using similar triangles and proportions. This problem involves geometric concepts, proportionality, and algebraic manipulation to find the unknown variable. Suitable for students in Grades 9-11.

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