Question

In the given figure, \( DE \) is drawn parallel to base \( BC \) of triangle \( ABC \). If \( DB = 7.2 \) cm, \( AE = 1.8 \) cm, and \( EC = 5.4 \) cm, then the value of \( AD \) will be:

• A. 2 cm
• B. 2.1 cm
• C. 2.4 cm
• D. 2.5 cm

Hint:

In similar triangles, the corresponding sides are proportional. Use the Basic Proportionality Theorem (Thales’ Theorem) to solve problems with parallel lines in triangles.

Answer 1

The Correct Option is A

We are given that \( DE \parallel BC \), which means triangles \( ADE \) and \( ABC \) are similar by the Basic Proportionality Theorem (or Thales’ Theorem). By the theorem, we have the following proportion: \[ \frac{AD}{AB} = \frac{AE}{AC} \]
Step 1: Calculate \( AB \) and \( AC \).
We know: - \( DB = 7.2 \) cm - \( EC = 5.4 \) cm - \( AE = 1.8 \) cm Thus, \( AB = AD + DB \) and \( AC = AE + EC \). So, \( AB = AD + 7.2 \) \( AC = 1.8 + 5.4 = 7.2 \)
Step 2: Set up the proportion.
From the similarity of the triangles: \[ \frac{AD}{AB} = \frac{AE}{AC} = \frac{1.8}{7.2} \] \[ \frac{AD}{AD + 7.2} = \frac{1.8}{7.2} \]
Step 3: Solve for \( AD \).
Cross-multiply to solve for \( AD \): \[ AD \times 7.2 = (AD + 7.2) \times 1.8 \] \[ 7.2 \, AD = 1.8 \, AD + 12.96 \] \[ 7.2 \, AD - 1.8 \, AD = 12.96 \] \[ 5.4 \, AD = 12.96 \] \[ AD = \frac{12.96}{5.4} = 2 \, \text{cm} \]
Step 4: Conclusion.
Thus, the value of \( AD \) is \( 2 \) cm.