In `Delta ABC`, DE || BC. If AD = 2x - 1, AE = 2x + 5, BD = x - 3 and CE = x - 1, then the value of x is:

Options

A. 8
B. 9
C. 10
D. 11

Answer

To solve the problem, we will use the Basic Proportionality Theorem (also known as Thales' theorem), 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. **Set Up the Proportionality Equation**:
Given that \( DE \parallel BC \), we can set up the following proportion based on the segments created by the parallel line:
\[ \frac{AD}{DB} = \frac{AE}{EC} \]

2. **Substitute the Given Values**:
We know:
- \( AD = 2x - 1 \)
- \( DB = x - 3 \)
- \( AE = 2x + 5 \)
- \( EC = x - 1 \)

Substituting these values into the proportion gives:
\[ \frac{2x - 1}{x - 3} = \frac{2x + 5}{x - 1} \]

3. **Cross-Multiply to Eliminate the Fractions**:
Cross-multiplying leads to:
\[ (2x - 1)(x - 1) = (2x + 5)(x - 3) \]

4. **Expand Both Sides**:
Expanding the left-hand side:
\[ 2x^2 - 2x - x + 1 = 2x^2 - 3x + 1 \]
Expanding the right-hand side:
\[ 2x^2 - 6x + 5x - 15 = 2x^2 - x - 15 \]

5. **Set the Equation**:
Now we have:
\[ 2x^2 - 3x + 1 = 2x^2 - x - 15 \]

6. **Simplify the Equation**:
Subtract \( 2x^2 \) from both sides:
\[ -3x + 1 = -x - 15 \]
Rearranging gives:
\[ -3x + x = -15 - 1 \]
Simplifying further:
\[ -2x = -16 \]

7. **Solve for x**:
Dividing both sides by -2:
\[ x = 8 \]

### Final Answer:
The value of \( x \) is \( 8 \).

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