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Let's solve the second part of Question 9, which is to prove the following:

\[
\frac{CA}{PA} = \frac{BC}{MP}
\]

### Problem Setup:

We are given that \( \triangle ABC \) and \( \triangle AMP \) are right-angled triangles. \( \triangle ABC \) is right-angled at \( B \), and \( \triangle AMP \) is right-angled at \( M \). From part (i), we already know that these two triangles are **similar** by the AA similarity criterion.

### Solution Steps:

1. **Using the similarity from part (i)**:
   Since \( \triangle ABC \sim \triangle AMP \) (proven in part i), the corresponding sides of the triangles will be in proportion. 

2. **Corresponding sides**:
   From the similarity, the corresponding sides of the two triangles are:
   - \( CA \) (the hypotenuse of \( \triangle ABC \)) corresponds to \( PA \) (the hypotenuse of \( \triangle AMP \)).
   - \( BC \) (the side opposite \( \angle A \) in \( \triangle ABC \)) corresponds to \( MP \) (the side opposite \( \angle A \) in \( \triangle AMP \)).

3. **Proportionality of corresponding sides**:
   Since the triangles are similar, the ratio of the corresponding sides will be equal:
   \[
   \frac{CA}{PA} = \frac{BC}{MP}
   \]

Thus, the given equation is proven based on the similarity of the two triangles.

---

Would you like to proceed with another part of this problem or discuss further details?

Here are some related questions for you:

1. How can we use Pythagoras' theorem to calculate the unknown sides in right-angled triangles?
2. What happens to the ratios of corresponding sides when the triangles are congruent instead of similar?
3. How can similarity in triangles be used to solve real-world measurement problems?
4. What other properties of the triangles can you infer once the similarity is established?
5. How does the AA similarity criterion differ from the SSS and SAS criteria?

**Tip:** In right-angled triangles, always look for the possibility of similarity through AA when one common angle and one right angle are given.

In this geometry problem, learn how to prove the similarity of two right-angled triangles ABC and AMP using the AA similarity criterion and demonstrate the proportionality of corresponding sides, CA/PA = BC/MP. Suitable for Grade 10-11 students.

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