P and Q are the points on the sides DE and DF of a triangle DEF such that DP = 5 cm, DE = 15 cm, DQ = 6 cm and QF = 18 cm. Is PQ||EF? Give reasons for your answer

Answer

To determine whether line segment PQ is parallel to line segment EF in triangle DEF, we will use the Basic Proportionality Theorem (also known as Thales' theorem). According to this theorem, 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 Measurements:**
- DP = 5 cm
- DE = 15 cm
- DQ = 6 cm
- QF = 18 cm

2. **Calculate the Remaining Lengths:**
- Since DE = 15 cm and DP = 5 cm, we can find PE (the remaining length):
\[ PE = DE - DP = 15 \, \text{cm} - 5 \, \text{cm} = 10 \, \text{cm} \]

3. **Set Up the Proportions:**
- According to the Basic Proportionality Theorem, if PQ is parallel to EF, then:
\[ \frac{DQ}{QF} = \frac{DP}{PE} \]

4. **Substitute the Values:**
- Substitute the known values into the proportion:
\[ \frac{DQ}{QF} = \frac{6}{18} \quad \text{and} \quad \frac{DP}{PE} = \frac{5}{10} \]

5. **Simplify the Ratios:**
- Simplifying the left side:
\[ \frac{6}{18} = \frac{1}{3} \]
- Simplifying the right side:
\[ \frac{5}{10} = \frac{1}{2} \]

6. **Compare the Ratios:**
- Now we compare the two ratios:
\[ \frac{1}{3} \neq \frac{1}{2} \]

7. **Conclusion:**
- Since the ratios are not equal, we conclude that:
\[ PQ \text{ is not parallel to } EF. \]