P and Q are points on sides AB and AC respectively of $ riangle ABC$. If $AP=3$ cm, $PB=6$ cm, $AQ=5$ cm and $QC=10$ cm, then:

Q: P and Q are points on sides AB and AC respectively of $\triangle ABC$. If $AP=3$ cm, $PB=6$ cm, $AQ=5$ cm and $QC=10$ cm, then:

P and Q are points on sides AB and AC respectively of $\triangle ABC$. If $AP=3$ cm, $PB=6$ cm, $AQ=5$ cm and $QC=10$ cm, then:

Question

P and Q are points on sides AB and AC respectively of $	riangle ABC$. If $AP=3$ cm, $PB=6$ cm, $AQ=5$ cm and $QC=10$ cm, then:

Accepted Answer

To determine why Option 2 ($BC = 3~PQ$) is the correct answer, we need to analyze the given triangle and the segments involved.

### Explanation:

We are given a triangle $	riangle ABC$ with points $P$ and $Q$ on sides $AB$ and $AC$, respectively. The lengths are as follows:
- $AP = 3$ cm, $PB = 6$ cm, so $AB = AP + PB = 9$ cm.
- $AQ = 5$ cm, $QC = 10$ cm, so $AC = AQ + QC = 15$ cm.

The problem involves finding the relationship between $BC$ and $PQ$. To solve this, we can use the concept of similar triangles or the intercept theorem (also known as Thales' theorem).

#### Step-by-step Justification:

1. **Proportional Segments:**
   - Since $P$ and $Q$ divide $AB$ and $AC$ in the ratios $AP:PB = 1:2$ and $AQ:QC = 1:2$, respectively, the segments $PQ$ and $BC$ are parallel. This is a consequence of the intercept theorem, which states that if a line divides two sides of a triangle proportionally, it is parallel to the third side.

2. **Similarity of Triangles:**
   - The line segment $PQ$ is parallel to $BC$, which implies that $	riangle APQ \sim 	riangle ABC$ by the Basic Proportionality Theorem (or Thales' theorem).

3. **Ratio of Similarity:**
   - Since $AP:PB = 1:2$, the ratio of similarity between $	riangle APQ$ and $	riangle ABC$ is $1:3$. This means that every corresponding length in $	riangle APQ$ is $\frac{1}{3}$ of the corresponding length in $	riangle ABC$.

4. **Calculating $PQ$:**
   - Since $PQ$ is parallel to $BC$ and $	riangle APQ \sim 	riangle ABC$, the length of $PQ$ is $\frac{1}{3}$ of the length of $BC$.

5. **Conclusion:**
   - Therefore, $BC = 3 	imes PQ$, which matches Option 2.

### Why Other Options are Incorrect:

- **Option 1 ($BC = 2~PQ$):** This option suggests that $BC$ is twice the length of $PQ$. However, based on the similarity ratio derived from the given segment ratios, $BC$ is actually three times the length of $PQ$, not twice.

- **Option 3 ($BC = 4~PQ$):** This option implies that $BC$ is four times the length of $PQ$. This is incorrect because the similarity ratio between $	riangle APQ$ and $	riangle ABC$ is $1:3$, not $1:4$.

- **Option 4 ($BC = 5~PQ$):** This option suggests that $BC$ is five times the length of $PQ$. This is incorrect as the similarity ratio clearly indicates a $1:3$ relationship, not $1:5$.

In conclusion, the correct relationship between $BC$ and $PQ$ is that $BC = 3~PQ$, which is why Option 2 is the correct answer.