Question

If in a \(\Delta ABC\), \(AB = 6\) cm and \(DE \parallel BC\) such that \(AE = \frac{1}{3} AC\), then the length of BD is

• A. 2 cm
• B. 3 cm
• C. 4 cm
• D. 5 cm

Hint:

An alternative way to use BPT directly:
If \(AE = \frac{1}{3} AC\), then \(EC = \frac{2}{3} AC\).
Thus, \(\frac{AD}{DB} = \frac{AE}{EC} = \frac{1}{2} \implies DB = 2 AD\).
Since \(AD + DB = 6 \implies AD + 2AD = 6 \implies AD = 2 \text{ cm}, DB = 4 \text{ cm}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given a triangle \(ABC\) where a line segment \(DE\) is parallel to \(BC\) with \(D\) on \(AB\) and \(E\) on \(AC\). We need to find the length of segment \(BD\) given that \(AB = 6 \text{ cm}\) and \(AE = \frac{1}{3} AC\).

Step 2: Key Formula or Approach:
According to Thales's Theorem (Basic Proportionality Theorem):
If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]

Step 3: Detailed Explanation:

• Given:
\[ AE = \frac{1}{3} AC \implies \frac{AE}{AC} = \frac{1}{3} \]

• Using BPT:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]
Substitute the known values:
\[ \frac{AD}{6} = \frac{1}{3} \]

• Solve for \(AD\):
\[ AD = \frac{6}{3} = 2 \text{ cm} \]

• Calculate the length of segment \(BD\):
\[ BD = AB - AD \]
\[ BD = 6 - 2 = 4 \text{ cm} \]


Step 4: Final Answer:
The length of \(BD\) is 4 cm.