Question

In the adjoining figure, \(AP = \frac{1}{2} AB\) and \(PQ \parallel BC\). If \(CQ = 3\) cm, then find the length of AC.

Hint:

If \(\frac{AP}{AB} = \frac{1}{2}\), it means \(AP = PB\). The ratio of the segments is \(1:1\).

Answer 1

Step 1: Understanding the Concept:
By the Basic Proportionality Theorem (BPT), if a line is parallel to one side of a triangle, it divides the other two sides proportionally.
Step 2: Key Formula or Approach:
If \(PQ \parallel BC\), then \(\frac{AP}{AB} = \frac{AQ}{AC}\).
Step 3: Detailed Explanation:
1. Given: \(AP = \frac{1}{2} AB \implies \frac{AP}{AB} = \frac{1}{2}\).
2. This implies \(P\) is the midpoint of \(AB\). Since \(PQ \parallel BC\), by the converse of midpoint theorem or BPT, \(Q\) must be the midpoint of \(AC\).
3. Therefore, \(AQ = QC\).
4. Given \(CQ = 3\) cm, then \(AQ = 3\) cm.
5. \(AC = AQ + QC = 3 + 3 = 6\) cm.
Step 4: Final Answer:
The length of \(AC\) is 6 cm.