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Let's work through the problems step by step:

### Question 8:
In \(\Delta ABC\), we are given:
- \( AB = 6 \, \text{cm} \)
- \( DE \parallel BC \)
- \( AE = \frac{1}{4} AC \)

We are asked to find the length of \( AD \).

#### Approach:
Since \( DE \parallel BC \) and \( AE = \frac{1}{4} AC \), we can apply the Basic Proportionality Theorem (Thales' Theorem), which tells us that the ratio of corresponding sides will be equal.

So, using the fact that \( \frac{AD}{DB} = \frac{AE}{EC} \) and knowing \( AE = \frac{1}{4}AC \), we can solve for \( AD \) in the following steps:
1. Let \( AC = x \). Then, \( AE = \frac{1}{4}x \).
2. This makes \( EC = \frac{3}{4}x \).
3. Since \( AD = \frac{1}{4}AB \), we can solve for \( AD \).

Thus, \( AD = 1.5 \, \text{cm} \).

**Answer for Question 8:** Option (c) \( 1.5 \, \text{cm} \).

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### Question 9:
In this question, we are given:
- \( DE \parallel BC \)
- \( \frac{AD}{DB} = \frac{2}{3} \)
- \( AC = 18 \, \text{cm} \)

We are asked to find the length of \( AE \).

#### Approach:
We will use the section formula based on the given ratio.

1. Since \( \frac{AD}{DB} = \frac{2}{3} \), we can express \( AD \) and \( DB \) in terms of \( AC \). 
   Let \( AD = 2x \) and \( DB = 3x \).
2. Therefore, \( AC = AD + DB = 2x + 3x = 5x \).
3. We are told that \( AC = 18 \, \text{cm} \), so \( 5x = 18 \).
4. Solving for \( x \), we get \( x = \frac{18}{5} = 3.6 \, \text{cm} \).
5. Thus, \( AD = 2x = 2 \times 3.6 = 7.2 \, \text{cm} \).

So, \( AE = AD = 7.2 \, \text{cm} \).

**Answer for Question 9:** Option (b) \( 7.2 \, \text{cm} \).

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Would you like further clarifications or details on these solutions?

Here are 5 related questions to explore:
1. What is the Basic Proportionality Theorem (Thales' Theorem)?
2. How does the parallel nature of lines in a triangle affect the ratio of the sides?
3. What is the significance of the section formula in geometry?
4. How do we determine the length of a side using a ratio in a triangle?
5. Can you derive the section formula for an arbitrary line dividing a triangle?

**Tip:** Always draw a diagram for problems involving ratios and proportions in triangles—it makes understanding relationships between sides much easier.

8. In ΔABC, AB = 6 cm and DE ∥ BC such that AE = 1/4 AC, find the length of AD. 9. In the given figure, DE is parallel to BC. If AD/DB = 2/3 and AC = 18 cm, find AE.

Learn how to solve triangle ratio problems involving Thales' Theorem and the section formula. The solution focuses on the proportionality of sides when parallel lines are present in triangles, including step-by-step calculations.

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