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: Use the whole-side form of the Basic Proportionality Theorem.
Since $DE\parallel BC$, $\frac{AE}{AC}=\frac{AD}{AB}$ directly, comparing each divided part to the whole side.
Step 2: Convert the given ratio into this form.
$\frac{AD}{AB}=\frac{AD}{AD+DB}=\frac{1}{1+3}=\frac14$ (using $\frac{AD}{DB}=\frac13$).
Step 3: Solve for AE without setting up a separate equation.
$\frac{AE}{6}=\frac14 \implies AE=\frac64=1.5$ cm, matching option (A).
\[ \boxed{1.5\text{ cm}} \]