Question

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

Hint:

When cross-multiplying in BPT problems, look for algebraic identities like \((x+2)(x-2) = x^2-4\) to simplify your work.

Answer 1

Step 1: Understanding the Concept:
By the Basic Proportionality Theorem (Thales' Theorem), if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Step 2: Key Formula or Approach:
\[ \frac{AD}{DB} = \frac{AE}{EC} \]
Step 3: Detailed Explanation:
1. Substitute the given values into the BPT ratio: \[ \frac{x}{x - 2} = \frac{x + 2}{x - 1} \] 2. Cross-multiply to solve the equation: \[ x(x - 1) = (x + 2)(x - 2) \] 3. Expand both sides (use \((a+b)(a-b) = a^2 - b^2\) for the right side): \[ x^2 - x = x^2 - 4 \] 4. Subtract \(x^2\) from both sides: \[ -x = -4 \] \[ x = 4 \]
Step 4: Final Answer:
The value of \(x\) is 4.