Question

Solve the following sub-questions (any four): In \( \triangle ABC \), \( DE \parallel BC \). If \( DB = 5.4 \, \text{cm} \), \( AD = 1.8 \, \text{cm} \), \( EC = 7.2 \, \text{cm} \), then find \( AE \).

Hint:

When applying the Basic Proportionality Theorem, ensure the line segment (DE) is parallel to the third side (BC). Always set up the ratio of the parts of one side equal to the ratio of the corresponding parts of the other side.

Answer 1

Step 1: Understanding the Concept:
The Basic Proportionality Theorem (BPT), also known as Thales's Theorem, states that if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.

Step 2: Key Formula or Approach:
According to the Basic Proportionality Theorem, since DE \(||\) BC, we have: \[ \frac{AD}{DB} = \frac{AE}{EC} \]

Step 3: Detailed Explanation:
Given: \[\begin{array}{rl} \bullet & \text{In \(\triangle\) ABC, DE \(||\) BC.} \\ \bullet & \text{AD = 1.8 cm} \\ \bullet & \text{DB = 5.4 cm} \\ \bullet & \text{EC = 7.2 cm} \\ \end{array}\] We need to find the length of AE.
Using the BPT, we set up the proportion: \[ \frac{AD}{DB} = \frac{AE}{EC} \] Substitute the given values into the equation: \[ \frac{1.8}{5.4} = \frac{AE}{7.2} \] First, simplify the ratio on the left side: \[ \frac{1.8}{5.4} = \frac{18}{54} = \frac{1}{3} \] Now the equation becomes: \[ \frac{1}{3} = \frac{AE}{7.2} \] To solve for AE, multiply both sides by 7.2: \[ AE = \frac{1}{3} \times 7.2 \] \[ AE = 2.4 \text{ cm} \]

Step 4: Final Answer:
The length of AE is 2.4 cm.