$ ext{In } riangle ABC, ext{ D is the midpoint of BC and E is the midpoint of AD. Then the area of } riangle BED ext{ is equal to :}$

Q: $\text{In } \triangle ABC, \text{ D is the midpoint of BC and E is the midpoint of AD. Then the area of } \triangle BED \text{ is equal to :}$

$\text{In } \triangle ABC, \text{ D is the midpoint of BC and E is the midpoint of AD. Then the area of } \triangle BED \text{ is equal to :}$

Question

$	ext{In } 	riangle ABC, 	ext{ D is the midpoint of BC and E is the midpoint of AD. Then the area of } 	riangle BED 	ext{ is equal to :}$

Accepted Answer

To determine why Option 1 is the correct answer, we need to analyze the given problem using geometric properties and reasoning.

### Explanation:

1. **Understanding the Problem:**
   - We have a triangle $ \triangle ABC $.
   - $ D $ is the midpoint of $ BC $, which means $ BD = DC $.
   - $ E $ is the midpoint of $ AD $, which means $ AE = ED $.

2. **Concept of Midpoints:**
   - Since $ D $ is the midpoint of $ BC $, line $ AD $ is a median of $ \triangle ABC $.
   - Since $ E $ is the midpoint of $ AD $, line $ BE $ is a median of $ \triangle ABD $.

3. **Area Calculation:**
   - The median of a triangle divides it into two smaller triangles of equal area. Therefore, $ \text{Area of } \triangle ABD = \frac{1}{2} \times \text{Area of } \triangle ABC $.
   - Similarly, since $ E $ is the midpoint of $ AD $, $ \triangle BED $ is half of $ \triangle ABD $.

4. **Area of $ \triangle BED $:**
   - Since $ \triangle BED $ is half of $ \triangle ABD $, the area of $ \triangle BED $ is:
     $$
     \text{Area of } \triangle BED = \frac{1}{2} \times \text{Area of } \triangle ABD = \frac{1}{2} \times \frac{1}{2} \times \text{Area of } \triangle ABC = \frac{1}{4} \times \text{Area of } \triangle ABC
     $$

5. **Conclusion:**
   - Therefore, the area of $ \triangle BED $ is indeed $\frac{1}{4} \times \text{Area of } \triangle ABC$, which corresponds to Option 1.

### Why Other Options are Incorrect:

- **Option 2 ($\frac{1}{3} \times \text{Area of } \triangle ABC$):**
  - This option suggests that $ \triangle BED $ is one-third of $ \triangle ABC $, which is incorrect because the division by midpoints results in a quarter, not a third.

- **Option 3 ($\frac{1}{5} \times \text{Area of } \triangle ABC$):**
  - This option implies a division into fifths, which does not align with the geometric properties of midpoints and medians in this context.

- **Option 4 ($\frac{2}{5} \times \text{Area of } \triangle ABC$):**
  - This option suggests a more complex division that does not result from the simple midpoint and median properties used in this problem.

In conclusion, the correct answer is Option 1, as the area of $ \triangle BED $ is $\frac{1}{4} \times \text{Area of } \triangle ABC$, based on the properties of midpoints and medians in a triangle.