In `DeltaABC` , if DE||BC , AD =x , DB =x-2,AE=x+2andEC=x-1 , then the value of x is

Answer

To solve the problem, we will use the Basic Proportionality Theorem (BPT) which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

### Step-by-Step Solution:

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

2. **Apply the Basic Proportionality Theorem**:
According to BPT, since \( DE \parallel BC \), we have:
\[ \frac{AD}{DB} = \frac{AE}{EC} \]

3. **Substitute the Values**:
Substitute the values of \( AD \), \( DB \), \( AE \), and \( EC \) into the equation:
\[ \frac{x}{x - 2} = \frac{x + 2}{x - 1} \]

4. **Cross Multiply**:
Cross multiplying gives us:
\[ x \cdot (x - 1) = (x - 2) \cdot (x + 2) \]

5. **Expand Both Sides**:
Expanding both sides:
- Left side: \( x^2 - x \)
- Right side: Using the difference of squares, \( (x - 2)(x + 2) = x^2 - 4 \)
So, we have:
\[ x^2 - x = x^2 - 4 \]

6. **Simplify the Equation**:
Subtract \( x^2 \) from both sides:
\[ -x = -4 \]

7. **Solve for \( x \)**:
Multiply both sides by -1:
\[ x = 4 \]

### Final Answer:
The value of \( x \) is \( 4 \).

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