https://file.aiquickdraw.com/mathbot/file/1730138983266r5fuszh3.jpg

Let's address each problem in turn based on the image provided.

### Problem 19
We need to write a paragraph proof for Theorem 9.2.

#### Given:
1. Triangle \( \triangle ADC \) is a right triangle.
2. \( DB \) is an altitude of \( \triangle ADC \).

#### Prove:
\[
\frac{AB}{DB} = \frac{DB}{CB}
\]

#### Proof:
Since \( DB \) is an altitude in \( \triangle ADC \), it creates two smaller right triangles, \( \triangle ADB \) and \( \triangle CDB \), each similar to the larger \( \triangle ADC \) by AA similarity (they each have a right angle and share another angle with \( \triangle ADC \)). By similarity, we know that corresponding side ratios are equal. Thus:
\[
\frac{AB}{DB} = \frac{DB}{CB}
\]
This completes the proof.

---

### Problem 20
This requires a two-column proof for Theorem 9.3.

#### Given:
1. \( \angle ADC \) is a right angle.
2. \( DB \) is an altitude of \( \triangle ADC \).

#### Prove:
\[
\frac{AB}{AD} = \frac{AD}{AC}, \quad \frac{BC}{DC} = \frac{DC}{AC}
\]

#### Two-Column Proof:
| Statements | Reasons |
|------------|---------|
| 1. \( \angle ADC \) is a right angle, \( DB \) is an altitude. | Given |
| 2. \( \triangle ADB \sim \triangle ADC \) and \( \triangle CDB \sim \triangle ADC \) | AA Similarity |
| 3. \( \frac{AB}{AD} = \frac{AD}{AC} \) | Corresponding side ratios of similar triangles |
| 4. \( \frac{BC}{DC} = \frac{DC}{AC} \) | Corresponding side ratios of similar triangles |

This completes the proof.

---

### Problem 21
We are asked to find the values of \( x \), \( y \), and \( z \) in the triangle.

#### Solution:
This involves using the Pythagorean theorem or similarity ratios, but we don’t have enough specific measurements in this part of the image to fully solve. Let me know if you have more details on side lengths or angles for a clearer solution.

---

### Problem 22
We are to check the work of Aiden and Tia regarding finding the value of \( x \).

1. **Aiden's Calculation**:
   - Set up the proportion \( \frac{3.6}{x} = \frac{x}{6} \).
   - Cross-multiplying, we get \( x^2 = 3.6 \times 6 \Rightarrow x^2 = 21.6 \Rightarrow x = \sqrt{21.6} \approx 4.6 \).
   - **Conclusion**: Aiden’s solution is correct.

2. **Tia's Calculation**:
   - Set up the proportion \( \frac{3.6}{x} = \frac{x}{8} \).
   - Cross-multiplying, we get \( x^2 = 3.6 \times 8 \Rightarrow x^2 = 28.8 \Rightarrow x = \sqrt{28.8} \approx 5.4 \).
   - **Conclusion**: Tia’s solution is incorrect as she used the wrong proportion.

---

Would you like further details on any part of these solutions? 

### Related Questions
1. How can similarity be used to prove more complex geometric relationships?
2. Why does the altitude in a right triangle create similar triangles?
3. Can we find unknown lengths without knowing angles in a triangle?
4. How can we verify the accuracy of proportions in geometric figures?
5. What methods can be used if we know only partial triangle side lengths?

### Tip
When dealing with similar triangles, always look for corresponding side ratios. They provide the foundation for most similarity-based proofs and calculations.

Solve problems 19, 20, 21, and 22 based on the provided triangle and given values. Prove certain relationships and find the values of x, y, and z.

Explore step-by-step solutions to problems involving right triangles, similarity theorems, and ratios. Learn how to prove relationships between triangle sides and find unknown values in geometric figures. Suitable for students in Grades 9-10.

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