In `DeltaABC, DE||BC `such that `(AD)/(BD) = 3/5`. If `AC = 5.6 cm`, then AE is equal to

Options

A. 4.2 cm.
B. 3.1 cm.
C. 2.8 cm.
D. 2.1 cm.

Answer

To solve the problem step by step, we will use the properties of similar triangles and the given ratio.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have triangle \( ABC \).
- Line \( DE \) is parallel to \( BC \).
- The ratio \( \frac{AD}{BD} = \frac{3}{5} \).
- The length \( AC = 5.6 \, \text{cm} \).

2. **Set Up the Ratios**:
- Since \( DE \) is parallel to \( BC \), by the Basic Proportionality Theorem (or Thales' theorem), we have:
\[ \frac{AD}{BD} = \frac{AE}{EC} \]
- Given \( \frac{AD}{BD} = \frac{3}{5} \), we can set:
\[ AD = 3x \quad \text{and} \quad BD = 5x \]
- Therefore, the total length of \( AB \) is:
\[ AB = AD + BD = 3x + 5x = 8x \]

3. **Calculate the Lengths**:
- Since \( AC \) is given as \( 5.6 \, \text{cm} \), we can express \( AC \) in terms of \( AE \) and \( EC \):
\[ AC = AE + EC \]
- From the ratio \( \frac{AE}{EC} = \frac{AD}{BD} = \frac{3}{5} \), we can set:
\[ AE = 3y \quad \text{and} \quad EC = 5y \]
- Therefore, the total length of \( AC \) becomes:
\[ AC = AE + EC = 3y + 5y = 8y \]

4. **Relate the Two Expressions for AC**:
- Since both expressions represent \( AC \):
\[ 8y = 5.6 \]
- To find \( y \):
\[ y = \frac{5.6}{8} = 0.7 \]

5. **Calculate AE**:
- Now substituting \( y \) back to find \( AE \):
\[ AE = 3y = 3 \times 0.7 = 2.1 \, \text{cm} \]

### Final Answer:
Thus, the length of \( AE \) is \( 2.1 \, \text{cm} \).

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