In figure-2, `DE||BC`. If `(AD)/(DB)=(3)/(2)` and `AE=2.7` cm then `EC` is equal to

Options

A. 2.0 cm
B. 1.8 cm
C. 4.0 cm
D. 2.7 cm

Answer

To solve the problem step by step, we will use the Basic Proportionality Theorem (BPT).

### Step-by-Step Solution:

1. **Understand the Given Information:**
- We have two segments, AD and DB, with a ratio of \( \frac{AD}{DB} = \frac{3}{2} \).
- The length of AE is given as 2.7 cm.
- We need to find the length of EC.

2. **Use the Basic Proportionality Theorem (BPT):**
- According to BPT, if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
- Therefore, we can write:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]
- Here, \( AB = AD + DB \). Since \( AD : DB = 3 : 2 \), we can express this as:
\[ AD = 3x \quad \text{and} \quad DB = 2x \]
- Thus, \( AB = AD + DB = 3x + 2x = 5x \).

3. **Substituting into the Proportionality Equation:**
- Now we can substitute into the BPT equation:
\[ \frac{AD}{AB} = \frac{3x}{5x} = \frac{3}{5} \]
- This gives us:
\[ \frac{3}{5} = \frac{AE}{AC} \]

4. **Substituting the Known Value of AE:**
- We know \( AE = 2.7 \) cm, so we substitute that into the equation:
\[ \frac{3}{5} = \frac{2.7}{AC} \]

5. **Cross-Multiplying to Solve for AC:**
- Cross-multiplying gives:
\[ 3 \cdot AC = 5 \cdot 2.7 \]
- Therefore:
\[ AC = \frac{5 \cdot 2.7}{3} \]
- Calculating this:
\[ AC = \frac{13.5}{3} = 4.5 \text{ cm} \]

6. **Finding EC:**
- We know that:
\[ AC = AE + EC \]
- Rearranging gives:
\[ EC = AC - AE \]
- Substituting the values we found:
\[ EC = 4.5 - 2.7 = 1.8 \text{ cm} \]

### Final Answer:
Thus, the length of EC is **1.8 cm**.