In `DeltaABC` DE||BC, where DE intersects AB and AC at the points D and E, respectively. If AD = 6 cm, DB = 12X- 6cm, AE= 2x cm and CE = 16 - 2x cm, then the value of X is

Options

A. 6 cm
B. 4 cm
C. 2 cm
D. 8 cm

Answer

To solve the problem, we will use the properties of similar triangles and the concept of proportionality. Given that DE is parallel to BC, we can set up a proportion based on the segments created by the intersection of DE with AB and AC.

### Step-by-Step Solution:

1. **Identify the segments**:
- We have AD = 6 cm
- DB = 12x - 6 cm
- AE = 2x cm
- CE = 16 - 2x cm

2. **Write the expressions for AB and AC**:
- The length of AB can be expressed as:
\[ AB = AD + DB = 6 + (12x - 6) = 12x \]
- The length of AC can be expressed as:
\[ AC = AE + CE = 2x + (16 - 2x) = 16 \]

3. **Set up the proportion**:
Since DE is parallel to BC, we can use the property of similar triangles:
\[ \frac{AD}{DB} = \frac{AE}{CE} \]
Substituting the values we have:
\[ \frac{6}{12x - 6} = \frac{2x}{16 - 2x} \]

4. **Cross-multiply to eliminate the fractions**:
\[ 6(16 - 2x) = 2x(12x - 6) \]
Expanding both sides:
\[ 96 - 12x = 24x^2 - 12x \]

5. **Rearranging the equation**:
Move all terms to one side:
\[ 24x^2 - 12x - 96 + 12x = 0 \]
Simplifying gives:
\[ 24x^2 - 96 = 0 \]

6. **Factor out the common term**:
\[ 24(x^2 - 4) = 0 \]
This simplifies to:
\[ x^2 - 4 = 0 \]

7. **Solve for x**:
\[ x^2 = 4 \]
Taking the square root of both sides gives:
\[ x = 2 \quad \text{or} \quad x = -2 \]
Since x represents a length, we only consider the positive value:
\[ x = 2 \]

### Final Answer:
The value of \( x \) is \( 2 \) cm.

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