In `DeltaABC` D and E are points on sides AB and AC, such that DE || BC. If AD = x,DB = x - 2, AE = x + 2 and EC = x-1 then the value of x is

Options

A. 4
B. 2
C. 1
D. 8

Answer

To solve the problem, we will use the properties of similar triangles. Given that \( DE \parallel BC \), we can establish a proportion between the segments of the sides of triangle \( ABC \) and triangle \( ADE \).

### Step-by-Step Solution:

1. **Identify the segments**:
- Let \( AD = x \)
- Let \( DB = x - 2 \)
- Let \( AE = x + 2 \)
- Let \( EC = x - 1 \)

2. **Calculate the lengths of sides \( AB \) and \( AC \)**:
- The length of side \( AB \) is:
\[ AB = AD + DB = x + (x - 2) = 2x - 2 \]
- The length of side \( AC \) is:
\[ AC = AE + EC = (x + 2) + (x - 1) = 2x + 1 \]

3. **Set up the proportion using the similarity of triangles**:
Since \( DE \parallel BC \), the triangles \( ADE \) and \( ABC \) are similar. Therefore, we can write:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]
Substituting the lengths we found:
\[ \frac{x}{2x - 2} = \frac{x + 2}{2x + 1} \]

4. **Cross-multiply to eliminate the fractions**:
\[ x(2x + 1) = (x + 2)(2x - 2) \]

5. **Expand both sides**:
- Left side:
\[ 2x^2 + x \]
- Right side:
\[ (x + 2)(2x - 2) = 2x^2 - 2x + 4x - 4 = 2x^2 + 2x - 4 \]

6. **Set the equation**:
\[ 2x^2 + x = 2x^2 + 2x - 4 \]

7. **Simplify the equation**:
Subtract \( 2x^2 \) from both sides:
\[ x = 2x - 4 \]
Rearranging gives:
\[ x - 2x = -4 \implies -x = -4 \implies x = 4 \]

8. **Conclusion**:
The value of \( x \) is \( 4 \).