Question

In the given ΔABC, if DE || BC, \(\frac{AD}{DB}=\frac{3}{5}\) and AC = 5.6 cm, then AE= ___.

• A. 2.1 cm
• B. 2 cm
• C. 2.2 cm
• D. 4.2 cm

Answer 1

The Correct Option is A

In triangle \( \triangle ABC \), given that \( DE \parallel BC \) and the ratio \( \frac{AD}{DB} = \frac{3}{5} \), we can use the Basic Proportionality Theorem (also known as Thales' Theorem). The theorem states that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. This implies that: \[ \frac{AE}{EC} = \frac{AD}{DB} = \frac{3}{5} \] Given that \( AC = 5.6 \) cm, we can write: \[ \frac{AE}{EC} = \frac{3}{5} \quad \text{and} \quad AE + EC = AC = 5.6 \, \text{cm} \] Let \( AE = x \) and \( EC = 5.6 - x \). Using the ratio \( \frac{x}{5.6 - x} = \frac{3}{5} \), we solve for \( x \): \[ 5x = 3(5.6 - x) \] \[ 5x = 16.8 - 3x \] \[ 5x + 3x = 16.8 \] \[ 8x = 16.8 \] \[ x = \frac{16.8}{8} = 2.1 \, \text{cm} \] Thus, \( AE = 2.1 \, \text{cm} \).

The correct option is (A): \(2.1\ cm\)