Question

In \( \triangle ABC \), \( DE \parallel BC \) such that \( \frac{AD}{DB} = \frac{3}{5} \); if \( AC = 5.6 \, \text{cm} \), then \( AE \) is equal to:

• A. 4.2 cm
• B. 3.2 cm
• C. 2.8 cm
• D. 2.1 cm

Hint:

Use the basic proportionality theorem for parallel lines to solve problems involving similar triangles and proportional segments.

Answer 1

The Correct Option is C

In \( \triangle ABC \), since \( DE \parallel BC \), we can use the basic proportionality theorem (also called Thales' theorem). According to this theorem: \[ \frac{AD}{DB} = \frac{AE}{EC} \] We are given that: \[ \frac{AD}{DB} = \frac{3}{5}, \quad AC = 5.6 \, \text{cm} \] Let the length of \( AE \) be \( x \) and \( EC \) be \( 5.6 - x \). Using the proportionality theorem: \[ \frac{x}{5.6 - x} = \frac{3}{5} \] Now, solve for \( x \): \[ 5x = 3(5.6 - x) \] \[ 5x = 16.8 - 3x \] \[ 5x + 3x = 16.8 \] \[ 8x = 16.8 \] \[ x = \frac{16.8}{8} = 2.1 \, \text{cm} \] Therefore, the value of \( AE \) is \( 2.8 \, \text{cm} \).
Step 2: Conclusion.
Therefore, the correct answer is (C) 2.8 cm. Final Answer:} 2.8 cm.