Question

In \( \triangle ABC \), \( DE \parallel BC \) such that \( \frac{AD}{DB} = \frac{3}{5} \). If \( AC = 5.6 \, \text{cm} \), then \( AE \)= ?

• A. 4.2 cm
• B. 3.1 cm
• C. 2.8 cm
• D. 2.1 cm

Hint:

When a line is parallel to one side of a triangle and intersects the other two sides, the segments of these sides created by the intersection are proportional to the segments of the corresponding sides.

Answer 1

The Correct Option is D

Step 1: Since \( DE \parallel BC \), the triangles \( \triangle ADE \) and \( \triangle ABC \) are similar by the Basic Proportionality Theorem. Step 2: Given \( \frac{AD}{DB} = \frac{3}{5} \), we calculate \( \frac{AD}{AB} \). Since \( AB = AD + DB \), let \( AD = 3x \) and \( DB = 5x \), so \( AB = 8x \). Therefore, \[ \frac{AD}{AB} = \frac{3x}{8x} = \frac{3}{8} \] Step 3: Using the similarity of the triangles, \[ \frac{AE}{AC} = \frac{AD}{AB} = \frac{3}{8} \] Given \( AC = 5.6 \, \text{cm} \), we calculate \( AE \) as follows: \[ AE = \frac{3}{8} \times 5.6 = 2.1 \, \text{cm} \] Thus, \( AE = 2.1 \) cm, and the correct answer is (D).