in triangle abc, de is parallel to bc. if ad = x, db = x-2, ae = x+2 and ec= x-1, find the value of x

Given triangle \( \triangle ABC \), with \( DE \parallel BC \), we can use the Basic Proportionality Theorem (also known as Thales' theorem). This theorem states that if a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides those sides in the same ratio.

### Step-by-step solution:

We are given:

- \( AD = x \)
- \( DB = x - 2 \)
- \( AE = x + 2 \)
- \( EC = x - 1 \)
  
Since \( DE \parallel BC \), the Basic Proportionality Theorem tells us:

\[
\frac{AD}{DB} = \frac{AE}{EC}
\]

Substitute the given values into this equation:

\[
\frac{x}{x - 2} = \frac{x + 2}{x - 1}
\]

### Step 1: Cross-multiply

\[
x(x - 1) = (x - 2)(x + 2)
\]

### Step 2: Expand both sides

Expanding both sides:

\[
x^2 - x = (x - 2)(x + 2)
\]
\[
x^2 - x = x^2 - 4
\]

### Step 3: Simplify the equation

Subtract \( x^2 \) from both sides:

\[
-x = -4
\]

### Step 4: Solve for \( x \)

Multiply both sides by -1:

\[
x = 4
\]

### Conclusion:
The value of \( x \) is \( \boxed{4} \).

Would you like any further clarification or details on this? 

Here are 5 related questions to explore further:
1. What is the Basic Proportionality Theorem (Thales' Theorem)?
2. How do parallel lines affect the properties of a triangle?
3. What happens when two sides of a triangle are divided in the same ratio?
4. Can the Basic Proportionality Theorem be applied to quadrilaterals?
5. How would the solution change if the triangle were isosceles?

**Tip:** Always check if lines are parallel in a triangle to apply the Basic Proportionality Theorem.

In triangle ABC, DE is parallel to BC. If AD = x, DB = x-2, AE = x+2 and EC = x-1, find the value of x.

Find the value of x in triangle ABC using Thales' Theorem. With DE parallel to BC, and given AD = x, DB = x-2, AE = x+2, and EC = x-1, apply the Basic Proportionality Theorem to solve this geometry problem. Suitable for students in Grades 9-10.

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