Question

In the given figure, \(∠ADE = ∠CBA\), if AD=3.8 cm, AE=3.6 cm, BE= 2.1 cm and BC= 4.2 cm, then DE =

• A. 2.8 cm
• B. 2.1 cm
• C. 3 cm
• D. 3.8 cm

Answer 1

The Correct Option is A

To solve the problem, we are given that \( \angle ADE = \angle CBA \), indicating that triangles \( \triangle ADE \) and \( \triangle CBA \) are similar by AA similarity.

1. Apply Triangle Similarity:
Since \( \triangle ADE \sim \triangle CBA \), the corresponding sides are proportional:

\[ \frac{DE}{BC} = \frac{AE}{AB} \]

2. Plug in the Known Values:
Given:
\( AE = 3.6 \, \text{cm}, \quad AB = AE + BE = 3.6 + 2.1 = 5.7 \, \text{cm} \)
\( BC = 4.2 \, \text{cm} \)

\[ \frac{DE}{4.2} = \frac{3.6}{5.7} \]

3. Solve for DE:
\[ DE = \frac{3.6}{5.7} \times 4.2 = \frac{36}{57} \times 4.2 = \frac{4}{6.33} \times 4.2 \approx 2.8 \, \text{cm} \]

Final Answer:
The length of \( DE \) is \({2.8 \, \text{cm}} \).