https://file.aiquickdraw.com/mathbot/file/1726768457092trmushgk.jpg

Let's solve question 9 from the provided image.

### Given:
In Fig. 6.39, \( \triangle ABC \) and \( \triangle AMP \) are two right-angled triangles, with right angles at \( B \) and \( M \), respectively.

We are tasked with proving two things:

#### (i) Prove that \( \triangle ABC \sim \triangle AMP \).
#### (ii) Prove that \( \frac{CA}{PA} = \frac{BC}{MP} \).

---

### (i) Proof: \( \triangle ABC \sim \triangle AMP \)

To prove the triangles are similar, we need to show that they have corresponding angles equal.

- In \( \triangle ABC \) and \( \triangle AMP \):
  - \( \angle ABC = \angle AMP = 90^\circ \) (Both triangles are right-angled at \( B \) and \( M \)).
  - \( \angle A \) is common to both triangles.

Thus, by **AA similarity** criterion (two angles are equal), the triangles are similar.

\[
\triangle ABC \sim \triangle AMP
\]

---

### (ii) Proof: \( \frac{CA}{PA} = \frac{BC}{MP} \)

Since we have already proven that \( \triangle ABC \sim \triangle AMP \), the corresponding sides of similar triangles are proportional.

Thus, from the similarity of \( \triangle ABC \) and \( \triangle AMP \), we can write the proportionality of corresponding sides as follows:

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

Hence, the required proportionality is proven:

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

---

Would you like a more detailed explanation or have any further questions? Here are some follow-up questions that may expand your understanding:

1. Why do we use the AA similarity criterion to prove the similarity of triangles?
2. What are some other criteria to prove similarity besides AA?
3. How does proportionality of sides follow from triangle similarity?
4. Can the concept of similar triangles be used in real-world applications?
5. How would the proof change if the triangles had different configurations?

**Tip:** When proving triangles are similar, always look for common angles and corresponding right angles first, as they provide an easy way to apply the AA criterion.

Solve the 9th question from the provided image: In Fig. 6.39, ABC and AMP are two right triangles, right-angled at B and M respectively. Prove that: (i) ΔABC ~ ΔAMP (ii) CA/PA = BC/MP.

This question explores the similarity of two right-angled triangles ΔABC and ΔAMP from Fig. 6.39, proving their similarity using the AA criterion and establishing the proportionality of their corresponding sides.

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