Question

In the given figure \(\triangle\)ABC is shown, in which DE \(\parallel\) BC. If AD = 5 cm, DB = 2.5 cm and DE = 8 cm, then the length of BC is :

• A. 10 cm
• B. 6 cm
• C. 12 cm
• D. 7.5 cm

Hint:

Avoid using \( \frac{AD}{DB} = \frac{DE}{BC} \). This is a common mistake. BPT relates segments of sides, but Similarity relates the full sides of the triangles.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In a triangle, if a line is parallel to one side, the smaller triangle formed is similar to the larger triangle (by AA similarity).
Step 2: Key Formula or Approach:
Similarity of triangles \( \triangle ADE \sim \triangle ABC \):
\[ \frac{AD}{AB} = \frac{DE}{BC} \]
Step 3: Detailed Explanation:
Given: \( AD = 5 \) cm, \( DB = 2.5 \) cm.
So, \( AB = AD + DB = 5 + 2.5 = 7.5 \) cm.
Using the similarity property:
\[ \frac{5}{7.5} = \frac{8}{BC} \]
Convert \( \frac{5}{7.5} \) to a simpler fraction: \( \frac{5 \times 10}{7.5 \times 10} = \frac{50}{75} = \frac{2}{3} \).
\[ \frac{2}{3} = \frac{8}{BC} \]
\[ 2 \times BC = 24 \]
\[ BC = 12 \text{ cm} \]
Step 4: Final Answer:
The length of BC is 12 cm.