https://file.aiquickdraw.com/mathbot/file/1736884231723mo55bum1.jpg

To solve for \( x \), let us analyze the diagram and the information provided:

### Key Observations:
1. **Given Triangle Properties**:
   - \( \triangle QRT \) and \( \triangle STU \) are right triangles.
   - \( RT = 9 \), \( QT = 10.8 \), and \( ST = 17.6 \).
   - \( x \) is the unknown side \( SU \).

2. **Relationship**:
   The triangles share a proportionality relationship because they are similar triangles. This similarity arises due to the presence of two right angles and proportional sides (as \( \angle R = \angle U \)).

### Step-by-Step Solution:
#### 1. Using Similar Triangles:
From similarity, we can write:
\[
\frac{ST}{QT} = \frac{SU}{RT}.
\]

#### 2. Substitute Known Values:
\[
\frac{17.6}{10.8} = \frac{x}{9}.
\]

#### 3. Solve for \( x \):
\[
x = \frac{9 \cdot 17.6}{10.8}.
\]

#### 4. Simplify:
First, calculate:
\[
9 \cdot 17.6 = 158.4.
\]
Then divide:
\[
x = \frac{158.4}{10.8} \approx 14.67.
\]

### Final Answer:
\[
x \approx 14.67
\]

Would you like a detailed explanation or have any follow-up questions?  

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### Related Questions:
1. How can we verify the similarity of triangles in this problem?
2. What is the proportionality constant between these triangles?
3. How can the Pythagorean theorem help verify the results?
4. Can you derive the other side lengths for each triangle?
5. How do similar triangle ratios work in general geometry problems?

### Tip:
Always cross-check proportionality ratios by verifying corresponding angles and sides!

Solve for x. Figures are not necessarily drawn to scale. Given two right triangles (triangle QRT and triangle STU) with dimensions QT = 10.8, RT = 9, ST = 17.6, and SU = x.

Learn how to solve for x in a geometry problem involving two similar right triangles. Explore key concepts like proportionality and the Triangle Similarity Theorem with a detailed step-by-step solution. Perfect for Grades 9-11 students.

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