If ABC is an equilateral triangle and P, Q, R respectively denote the middle points of AB, BC, CA then.

Options

A. `PQR` must be an equilateral triangle
B. `PQ + QR + PR = AB `
C. `PQ + QR + PR = 2 AB `
D. `PQR` must be a right angled triangle

Answer

To solve the problem, we need to analyze the properties of triangle ABC and the midpoints P, Q, and R.

### Step-by-Step Solution:

1. **Draw Triangle ABC**:
- Start by sketching an equilateral triangle ABC where all sides are equal. Let the length of each side be denoted as \( AB = BC = CA = x \).

2. **Identify Midpoints**:
- Mark the midpoints of each side:
- Let \( P \) be the midpoint of side \( AB \).
- Let \( Q \) be the midpoint of side \( BC \).
- Let \( R \) be the midpoint of side \( CA \).

3. **Determine Lengths of Segments**:
- Since \( P \), \( Q \), and \( R \) are midpoints:
- The length of \( AP = PB = \frac{x}{2} \)
- The length of \( BQ = QC = \frac{x}{2} \)
- The length of \( CR = RA = \frac{x}{2} \)

4. **Calculate Lengths of PQ, QR, and PR**:
- To find the lengths of segments \( PQ \), \( QR \), and \( PR \), we can use the properties of midpoints in a triangle:
- The segment \( PQ \) connects the midpoints of sides \( AB \) and \( BC \).
- By the midpoint theorem, \( PQ \) is parallel to \( AC \) and half its length, so \( PQ = \frac{1}{2} \times AC = \frac{x}{2} \).
- Similarly, \( QR = \frac{1}{2} \times AB = \frac{x}{2} \) and \( PR = \frac{1}{2} \times BC = \frac{x}{2} \).

5. **Sum of Lengths**:
- Now, calculate the total length of \( PQ + QR + PR \):
\[ PQ + QR + PR = \frac{x}{2} + \frac{x}{2} + \frac{x}{2} = \frac{3x}{2} \]

6. **Comparison with AB**:
- Since \( AB = x \), we can compare:
\[ PQ + QR + PR = \frac{3x}{2} \neq AB \]

7. **Conclusion**:
- Since \( PQR \) consists of segments that are equal in length and each side is half the length of the corresponding side of triangle ABC, triangle \( PQR \) is also equilateral.
- Therefore, the correct conclusion is that triangle \( PQR \) is an equilateral triangle.

### Final Answer:
The correct option is that \( PQR \) must be an equilateral triangle.