Question:
In \(\Delta DEF\), \(AB \parallel EF\). The value of \(x\) is :
In \(\Delta DEF\), \(AB \parallel EF\). The value of \(x\) is :
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Always check if your algebraic solutions are geometrically possible.
Since lengths of a triangle must be strictly positive, any value of \(x\) that makes a side length zero or negative must be discarded.
Since lengths of a triangle must be strictly positive, any value of \(x\) that makes a side length zero or negative must be discarded.
Updated On: Jul 7, 2026
- \(0, 2\)
- \(2\) only
- \(- 2\)
- \(1\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Question:
In the triangle \(\Delta DEF\), we are given a line segment \(AB\) which is parallel to side \(EF\).
The segment lengths are expressed in terms of \(x\):
- \(DA = 2x\)
- \(AE = 3x + 1\)
- \(DB = x\)
- \(BF = 2x - \frac{1}{2}\)
We need to determine the value of \(x\).
Step 2: Key Formula or Approach:
According to the Basic Proportionality Theorem (also known as Thales's Theorem), 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.
Therefore:
\[ \frac{DA}{AE} = \frac{DB}{BF} \]
Step 3: Detailed Explanation:
1. Write down the BPT ratio:
\[ \frac{2x}{3x + 1} = \frac{x}{2x - 1/2} \] 2. Since \(x\) represents a physical length, \(x\) cannot be \(0\). Thus, we can cancel \(x\) from the numerators on both sides:
\[ \frac{2}{3x + 1} = \frac{1}{2x - 1/2} \] 3. Cross-multiply to solve for \(x\):
\[ 2\left(2x - \frac{1}{2}\right) = 1(3x + 1) \] 4. Expand the terms on both sides:
\[ 4x - 1 = 3x + 1 \] 5. Rearrange the terms to isolate \(x\):
\[ 4x - 3x = 1 + 1 \] \[ x = 2 \] 6. If we do not cancel \(x\) initially, we would solve:
\[ 2x\left(2x - \frac{1}{2}\right) = x(3x + 1) \] \[ x(4x - 1) = x(3x + 1) \] \[ x(4x - 1 - 3x - 1) = 0 \] \[ x(x - 2) = 0 \implies x = 0 \text{ or } x = 2 \] 7. Since \(x = 0\) would make the lengths \(DA = 0\) and \(DB = 0\), which is geometrically impossible for a triangle, \(x = 0\) is rejected.
8. Thus, \(x = 2\) is the only valid solution.
Step 4: Final Answer:
The correct option is (B).
In the triangle \(\Delta DEF\), we are given a line segment \(AB\) which is parallel to side \(EF\).
The segment lengths are expressed in terms of \(x\):
- \(DA = 2x\)
- \(AE = 3x + 1\)
- \(DB = x\)
- \(BF = 2x - \frac{1}{2}\)
We need to determine the value of \(x\).
Step 2: Key Formula or Approach:
According to the Basic Proportionality Theorem (also known as Thales's Theorem), 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.
Therefore:
\[ \frac{DA}{AE} = \frac{DB}{BF} \]
Step 3: Detailed Explanation:
1. Write down the BPT ratio:
\[ \frac{2x}{3x + 1} = \frac{x}{2x - 1/2} \] 2. Since \(x\) represents a physical length, \(x\) cannot be \(0\). Thus, we can cancel \(x\) from the numerators on both sides:
\[ \frac{2}{3x + 1} = \frac{1}{2x - 1/2} \] 3. Cross-multiply to solve for \(x\):
\[ 2\left(2x - \frac{1}{2}\right) = 1(3x + 1) \] 4. Expand the terms on both sides:
\[ 4x - 1 = 3x + 1 \] 5. Rearrange the terms to isolate \(x\):
\[ 4x - 3x = 1 + 1 \] \[ x = 2 \] 6. If we do not cancel \(x\) initially, we would solve:
\[ 2x\left(2x - \frac{1}{2}\right) = x(3x + 1) \] \[ x(4x - 1) = x(3x + 1) \] \[ x(4x - 1 - 3x - 1) = 0 \] \[ x(x - 2) = 0 \implies x = 0 \text{ or } x = 2 \] 7. Since \(x = 0\) would make the lengths \(DA = 0\) and \(DB = 0\), which is geometrically impossible for a triangle, \(x = 0\) is rejected.
8. Thus, \(x = 2\) is the only valid solution.
Step 4: Final Answer:
The correct option is (B).
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