In `Delta ABC, D ` and E are the points on sides AB and AC respectively.
Find whether DE || BC, if :
AB = 6-3 cm, EC = 11:0 cm, AD = 0.8 cm and AE = 1.6 cm. 

Answer

To determine whether line segment DE is parallel to line segment BC in triangle ABC, we can use the properties of similar triangles and the converse of the midpoint theorem. Here’s the step-by-step solution:

### Step 1: Identify the given values
We have the following measurements:
- AB = 6.3 cm
- EC = 11 cm
- AD = 0.8 cm
- AE = 1.6 cm

### Step 2: Calculate the length of BD
Since D is a point on AB, we can find the length of BD using the formula:
\[ BD = AB - AD \]
Substituting the known values:
\[ BD = 6.3 \, \text{cm} - 0.8 \, \text{cm} = 5.5 \, \text{cm} \]

### Step 3: Set up the ratio of segments
We need to check the ratios of the segments:
\[ \frac{AE}{EC} \quad \text{and} \quad \frac{AD}{BD} \]

Calculating the ratio of AE to EC:
\[ \frac{AE}{EC} = \frac{1.6 \, \text{cm}}{11 \, \text{cm}} \]

Calculating the ratio of AD to BD:
\[ \frac{AD}{BD} = \frac{0.8 \, \text{cm}}{5.5 \, \text{cm}} \]

### Step 4: Simplify the ratios
Now we simplify both ratios:
1. For \( \frac{AE}{EC} \):
\[ \frac{1.6}{11} \]
This can be simplified to:
\[ \frac{1.6 \div 1.6}{11 \div 1.6} = \frac{1}{6.875} \]

2. For \( \frac{AD}{BD} \):
\[ \frac{0.8}{5.5} \]
This can be simplified to:
\[ \frac{0.8 \div 0.8}{5.5 \div 0.8} = \frac{1}{6.875} \]

### Step 5: Compare the ratios
Since both ratios are equal:
\[ \frac{AE}{EC} = \frac{AD}{BD} \]

### Step 6: Conclusion
By the converse of the midpoint theorem, if the ratios of the segments are equal, then DE is parallel to BC. Hence, we conclude that:
**DE is parallel to BC.**

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