Question

In $\Delta DEF$, AB $\parallel$ EF. The value of x is :

• A. $0, 2$
• B. $2$ only
• C. $-2$
• D. $1$

Hint:

In geometry problems, always double-check your solutions to ensure they don't produce zero or negative lengths.
If you solve the equation as a full quadratic:
$2x(2x - 0.5) = x(3x + 1) \implies 4x^2 - x = 3x^2 + x \implies x^2 - 2x = 0 \implies x(x-2) = 0$.
This gives two mathematical roots, $x = 0$ and $x = 2$.
However, $x = 0$ is rejected because side lengths in geometry must be strictly positive.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is based on the "Triangles" chapter, specifically focusing on similarity of triangles and the Basic Proportionality Theorem.
We are given a triangle $DEF$ where a line segment $AB$ is parallel to the side $EF$.
The line segment $AB$ intersects the other two sides, $DE$ and $DF$, at points $A$ and $B$, respectively.
We are given the lengths of the segments in terms of a variable $x$, and we need to determine the correct value of $x$.

Step 2: Key Formula or Approach:
We apply the

Basic Proportionality Theorem (Thales' Theorem), which states:
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.
For the given triangle $DEF$ with $AB \parallel EF$:
\[ \frac{DA}{AE} = \frac{DB}{BF} \]
We substitute the algebraic expressions for these segment lengths and solve the resulting equation for $x$.

Step 3: Detailed Explanation:

• Identify the lengths of the segments from the given figure:

• $DA = 2x$

• $AE = 3x + 1$

• $DB = x$

• $BF = 2x - \frac{1}{2} = \frac{4x - 1}{2}$

• According to the Basic Proportionality Theorem, write the ratio equation:
\[ \frac{DA}{AE} = \frac{DB}{BF} \]

• Substitute the segment values into the ratio:
\[ \frac{2x}{3x + 1} = \frac{x}{2x - 1/2} \]

• We can simplify this equation. Since $x$ represents a geometric length, $x$ cannot be equal to $0$ (a length of zero would make the triangle collapse).
Thus, we can divide both sides of the equation by $x$:
\[ \frac{2}{3x + 1} = \frac{1}{2x - 1/2} \]

• Cross-multiply to eliminate the fractions:
\[ 2 \left( 2x - \frac{1}{2} \right) = 1 \cdot (3x + 1) \]

• Expand both sides:
\[ 4x - 1 = 3x + 1 \]

• Rearrange the terms to isolate the variable $x$ on one side:
\[ 4x - 3x = 1 + 1 \] \[ x = 2 \]

• Let us verify if $x = 2$ yields positive lengths:

• $DA = 2(2) = 4 \gt 0$

• $AE = 3(2) + 1 = 7 \gt 0$

• $DB = 2 \gt 0$

• $BF = 2(2) - 0.5 = 3.5 \gt 0$
Since all geometric lengths are positive, $x = 2$ is a valid solution.


Step 4: Final Answer:
The only geometrically valid value of $x$ is $2$, which corresponds to Option (B).