Given an equilateral triangle ABC, D, E and F are the mid-points of the sides AB, BC and AC respectively, then the quadrilateral BEFD is exactly a

Options

A. square
B. rectangle
C. parallelogram
D. rhombus

Answer

To solve the problem, we need to analyze the given equilateral triangle ABC and the midpoints D, E, and F of its sides. We will determine the nature of the quadrilateral BEFD.

### Step-by-Step Solution:

1. **Identify the Triangle and Midpoints**:
- We have an equilateral triangle ABC.
- Let D be the midpoint of side AB, E be the midpoint of side BC, and F be the midpoint of side AC.

2. **Use the Midpoint Theorem**:
- According to the midpoint theorem, if D, E, and F are midpoints of sides AB, BC, and AC respectively, then:
- DE = 1/2 AC
- EF = 1/2 AB
- FD = 1/2 BC

3. **Establish Side Lengths**:
- Since ABC is an equilateral triangle, all sides are equal. Let the length of each side be 's'.
- Therefore, we have:
- DE = 1/2 * s
- EF = 1/2 * s
- FD = 1/2 * s

4. **Determine Lengths of Quadrilateral BEFD**:
- Now, we need to find the lengths of BE and BF:
- BE = 1/2 * AB = 1/2 * s
- BF = 1/2 * BC = 1/2 * s

5. **Compare the Sides of Quadrilateral BEFD**:
- Now we have:
- DE = EF = FD = BE = BF = 1/2 * s
- This means all sides of quadrilateral BEFD are equal.

6. **Conclusion**:
- Since all sides of quadrilateral BEFD are equal, it is a rhombus.

### Final Answer:
The quadrilateral BEFD is a rhombus.