A and Bare respectively the points on the sides PQ and PR of a triangle PQR such that PQ = 10.5 m PA = 4.5 m, BR= 8 m and PB= 6 m. Is `ABabs()QR`?

Answer

To determine if line segment AB is parallel to line segment QR in triangle PQR, we can use the Basic Proportionality Theorem (also known as Thales' theorem). This theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Length of side PQ = 10.5 m
- Length of segment PA = 4.5 m
- Length of segment PB = 6 m
- Length of segment BR = 8 m

2. **Calculate AQ:**
- Since PQ = PA + AQ, we can find AQ:
\[ AQ = PQ - PA = 10.5 \, \text{m} - 4.5 \, \text{m} = 6 \, \text{m} \]

3. **Calculate BR:**
- Since PR = PB + BR, we can find BR:
\[ BR = PR - PB = 8 \, \text{m} \]

4. **Set Up the Ratios:**
- Now, we can set up the ratios of the segments:
\[ \frac{PA}{AQ} = \frac{4.5}{6} \]
- Simplifying this ratio:
\[ \frac{PA}{AQ} = \frac{3}{4} \]

5. **Calculate the Ratio PB/BR:**
- Now, we can calculate the ratio of PB to BR:
\[ \frac{PB}{BR} = \frac{6}{8} = \frac{3}{4} \]

6. **Compare the Ratios:**
- Since \(\frac{PA}{AQ} = \frac{PB}{BR}\), we have:
\[ \frac{3}{4} = \frac{3}{4} \]

7. **Conclusion:**
- According to the Basic Proportionality Theorem, since the ratios are equal, we can conclude that line segment AB is parallel to line segment QR.

### Final Answer:
Yes, line segment AB is parallel to line segment QR.