Question

In \(\triangle ABC\), if \(DE \parallel BC\), \(AD = x\), \(DB = x - 2\), \(AE = x + 2\), and \(EC = x - 1\), then the value of \(x\) is:

• A. 3
• B. 2
• C. 1
• D. 4

Hint:

Use the Basic Proportionality Theorem (Thales' Theorem) when dealing with parallel lines in triangles. It helps relate the segments on two sides divided by a parallel line.

Answer 1

The Correct Option is B

Using the Basic Proportionality Theorem (Thales' Theorem), which states that if a line divides two sides of a triangle in the same ratio, the line is parallel to the third side, we have: \[ \frac{AD}{DB} = \frac{AE}{EC} \] Substituting the given values: \[ \frac{x}{x - 2} = \frac{x + 2}{x - 1} \] Now, cross-multiply to solve for \(x\): \[ x(x - 1) = (x - 2)(x + 2) \] Expanding both sides: \[ x^2 - x = x^2 - 4 \] Simplifying: \[ -x = -4 \quad \Rightarrow \quad x = 4 \] Thus, the correct answer is option (4).