We will solve this problem using the Basic Proportionality Theorem (Thales' Theorem), which states:

"If a line is drawn parallel to one side of a triangle, dividing the other two sides in the same ratio."

Given:

• $DE \parallel BC$
• $AD = 3$ cm
• $DE = 5$ cm
• $EC = 2$ cm
• We need to find $AE$.

Since $DE \parallel BC$, applying the Basic Proportionality Theorem,

$\frac{AD}{DB} = \frac{AE}{EC}$

Rearrange for $AE$:

$AE = \frac{AD}{DB} \times EC$

To find $DB$, observe that:

$DB = DE = 5 \text{ cm} \quad \text{(since DE is parallel to BC and proportional)}$

Now, substituting values:

$AE = \frac{3}{5} \times 2$

$AE = \frac{6}{5} = 1.2 \text{ cm}$

Final Answer:

$AE = 1.2 \text{ cm}$

Would you like a step-by-step diagram or more details?

Related Questions:

1. How does the Basic Proportionality Theorem apply to triangles?
2. If AD = 4 cm and EC = 3 cm with DE = 6 cm, what is AE?
3. What happens if DE is not parallel to BC? Would the theorem still apply?
4. Can you prove the Basic Proportionality Theorem using similar triangles?
5. What are real-life applications of Thales' Theorem?

Math Tip:

Whenever a line is drawn parallel to one side of a triangle, always check if you can apply the Basic Proportionality Theorem to simplify calculations! 🚀