Question

In the given figure, DE \(\parallel\) BC. If \(\frac{AD}{DB} = \frac{1}{3}\) and AC = 6 cm, then length AE is

• A. 1.5 cm
• B. 1 cm
• C. 2 cm
• D. 3 cm

Hint:

An alternative form of the Basic Proportionality Theorem relates a segment to the whole side:
\[ \frac{AE}{AC} = \frac{AD}{AB} = \frac{AD}{AD + DB} \] Substituting the ratio \(\frac{AD}{DB} = \frac{1}{3}\) directly gives:
\[ \frac{AE}{6} = \frac{1}{1 + 3} = \frac{1}{4} \implies AE = \frac{6}{4} = 1.5 \text{ cm} \] This avoids solving equations and is much faster!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The topic of this question is Triangles, specifically Thales' Theorem (Basic Proportionality 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.
We are given a triangle \(\Delta ABC\) where \(DE \parallel BC\).
We are given the ratio \(\frac{AD}{DB} = \frac{1}{3}\) and the total length \(AC = 6\) cm.
We need to determine the length of the segment \(AE\).

Step 2: Key Formula or Approach:
Since \(DE \parallel BC\), by the Basic Proportionality Theorem, we have:
\[ \frac{AE}{EC} = \frac{AD}{DB} \] Since \(AC = AE + EC = 6\) cm, we can write \(EC = 6 - AE\).
Substituting this into the ratio will allow us to solve for \(AE\).

Step 3: Detailed Explanation:

• Let the length of segment \(AE\) be \(x\) cm.

• Express \(EC\) in terms of \(x\):
Since the total length \(AC = 6\) cm:
\[ EC = AC - AE = 6 - x \]

• Apply Thales' Theorem:
\[ \frac{AE}{EC} = \frac{AD}{DB} \] \[ \frac{x}{6 - x} = \frac{1}{3} \]

• Cross-multiply to solve the linear equation:
\[ 3x = 1(6 - x) \] \[ 3x = 6 - x \] \[ 4x = 6 \] \[ x = \frac{6}{4} = 1.5 \text{ cm} \]

Step 4: Final Answer:
The length of segment \(AE\) is 1.5 cm.
Therefore, the correct option is (A).