Question:
In the given figure, \(DE \parallel BC\). If \(\frac{AD}{DB} = \frac{1}{3}\) and \(AC = 6\) cm, then length AE is
In the given figure, \(DE \parallel BC\). If \(\frac{AD}{DB} = \frac{1}{3}\) and \(AC = 6\) cm, then length AE is
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An alternative and faster form of the Basic Proportionality Theorem directly relates the segment to the entire side:
\[ \frac{AE}{AC} = \frac{AD}{AB} = \frac{AD}{AD + DB} \] Using this, we can write:
\[ \frac{AE}{6} = \frac{1}{1 + 3} = \frac{1}{4} \implies AE = \frac{6}{4} = 1.5 \text{ cm} \] This approach avoids having to define separate variables and is much quicker!
\[ \frac{AE}{AC} = \frac{AD}{AB} = \frac{AD}{AD + DB} \] Using this, we can write:
\[ \frac{AE}{6} = \frac{1}{1 + 3} = \frac{1}{4} \implies AE = \frac{6}{4} = 1.5 \text{ cm} \] This approach avoids having to define separate variables and is much quicker!
Updated On: Jul 9, 2026
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
The topic of this question is Triangles, specifically the Basic Proportionality Theorem (also known as Thales' Theorem).
The theorem states that 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.
In the given triangle \(\Delta ABC\), we are given that \(DE \parallel BC\).
We are given the ratio \(\frac{AD}{DB} = \frac{1}{3}\) and the total length of side \(AC = 6\) cm.
We need to determine the length of the segment \(AE\).
Step 2: Key Formula or Approach:
By the Basic Proportionality Theorem, since \(DE \parallel BC\) in \(\Delta ABC\), we have the following ratio:
\[ \frac{AE}{EC} = \frac{AD}{DB} \] We are given \(\frac{AD}{DB} = \frac{1}{3}\), so:
\[ \frac{AE}{EC} = \frac{1}{3} \] We also know that the total length \(AC\) is the sum of the segments \(AE\) and \(EC\):
\[ AC = AE + EC = 6 \text{ cm} \] We will express \(EC\) in terms of \(AE\) and substitute it into this relation to solve for \(AE\).
Step 3: Detailed Explanation:
• Let the length of the segment \(AE\) be represented by \(x\) cm.
• Express the length of segment \(EC\) in terms of \(x\):
Since the total length \(AC = AE + EC = 6\) cm:
\[ EC = AC - AE = 6 - x \]
• Write down the ratio from the Basic Proportionality Theorem:
\[ \frac{AE}{EC} = \frac{AD}{DB} \]
• Substitute the known values and expressions into the ratio:
\[ \frac{x}{6 - x} = \frac{1}{3} \]
• Cross-multiply to solve the linear equation:
\[ 3 \cdot x = 1 \cdot (6 - x) \] \[ 3x = 6 - x \]
• Group the \(x\) terms on the left-hand side:
\[ 3x + x = 6 \] \[ 4x = 6 \] \[ x = \frac{6}{4} \] \[ x = 1.5 \text{ cm} \]
• Therefore, the length of \(AE\) is \(1.5\) cm.
Step 4: Final Answer:
The length of segment \(AE\) is 1.5 cm.
Therefore, the correct option is (A).
The topic of this question is Triangles, specifically the Basic Proportionality Theorem (also known as Thales' Theorem).
The theorem states that 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.
In the given triangle \(\Delta ABC\), we are given that \(DE \parallel BC\).
We are given the ratio \(\frac{AD}{DB} = \frac{1}{3}\) and the total length of side \(AC = 6\) cm.
We need to determine the length of the segment \(AE\).
Step 2: Key Formula or Approach:
By the Basic Proportionality Theorem, since \(DE \parallel BC\) in \(\Delta ABC\), we have the following ratio:
\[ \frac{AE}{EC} = \frac{AD}{DB} \] We are given \(\frac{AD}{DB} = \frac{1}{3}\), so:
\[ \frac{AE}{EC} = \frac{1}{3} \] We also know that the total length \(AC\) is the sum of the segments \(AE\) and \(EC\):
\[ AC = AE + EC = 6 \text{ cm} \] We will express \(EC\) in terms of \(AE\) and substitute it into this relation to solve for \(AE\).
Step 3: Detailed Explanation:
• Let the length of the segment \(AE\) be represented by \(x\) cm.
• Express the length of segment \(EC\) in terms of \(x\):
Since the total length \(AC = AE + EC = 6\) cm:
\[ EC = AC - AE = 6 - x \]
• Write down the ratio from the Basic Proportionality Theorem:
\[ \frac{AE}{EC} = \frac{AD}{DB} \]
• Substitute the known values and expressions into the ratio:
\[ \frac{x}{6 - x} = \frac{1}{3} \]
• Cross-multiply to solve the linear equation:
\[ 3 \cdot x = 1 \cdot (6 - x) \] \[ 3x = 6 - x \]
• Group the \(x\) terms on the left-hand side:
\[ 3x + x = 6 \] \[ 4x = 6 \] \[ x = \frac{6}{4} \] \[ x = 1.5 \text{ cm} \]
• Therefore, the length of \(AE\) is \(1.5\) cm.
Step 4: Final Answer:
The length of segment \(AE\) is 1.5 cm.
Therefore, the correct option is (A).
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