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 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!

Answer 1

The Correct Option is A

Step 1: Identify similar triangles.
Since \(DE \parallel BC\) in \(\Delta ABC\), triangle \(ADE\) is similar to triangle \(ABC\) by the AA criterion, because \(\angle ADE = \angle ABC\) and \(\angle A\) is common.
Step 2: Write the ratio of corresponding sides.
From this similarity, \(\frac{AE}{AC} = \frac{AD}{AB}\).
Step 3: Convert the given ratio AD:DB into AD:AB.
Since \(\frac{AD}{DB} = \frac{1}{3}\), we have \(AD:DB = 1:3\), so \(AD:AB = 1:(1+3) = 1:4\).
Step 4: Solve for AE.
\[ \frac{AE}{AC} = \frac{1}{4} \implies AE = \frac{6}{4} = 1.5 \text{ cm} \]
This matches option (A).
\[ \boxed{AE = 1.5 \text{ cm}} \]