Let M,N, P and Q be the mid points of the edges AB, CD, AC and BD respectively of the tetrahedron ABCD. Further, MN is perpendicular to both AB and CD and PQ is perpendicular to both AC and BD. Then which of the following is/are correct:

Options

A. AB = CD
B. BC = DA
C. AC = BD
D. AN = BN

Answer

To solve the problem, we need to analyze the tetrahedron ABCD and the midpoints M, N, P, and Q of its edges. We will derive the relationships between the edges based on the given conditions.

### Step-by-step Solution:

1. **Identify the Midpoints**:
- Let \( M \) be the midpoint of edge \( AB \).
- Let \( N \) be the midpoint of edge \( CD \).
- Let \( P \) be the midpoint of edge \( AC \).
- Let \( Q \) be the midpoint of edge \( BD \).

2. **Establish the Perpendicular Conditions**:
- We are given that line segment \( MN \) is perpendicular to both edges \( AB \) and \( CD \).
- We are also given that line segment \( PQ \) is perpendicular to both edges \( AC \) and \( BD \).

3. **Analyze the Tetrahedron**:
- In a regular tetrahedron, all edges are of equal length. Thus, we can denote the lengths of the edges as follows:
- \( AB = a \)
- \( AC = b \)
- \( AD = c \)
- \( BC = d \)
- \( BD = e \)
- \( CD = f \)

4. **Use the Perpendicular Conditions**:
- Since \( MN \) is perpendicular to both \( AB \) and \( CD \), we can conclude that:
- \( AB \) is equal to \( CD \) (i.e., \( a = f \)).
- Similarly, since \( PQ \) is perpendicular to both \( AC \) and \( BD \), we can conclude that:
- \( AC \) is equal to \( BD \) (i.e., \( b = e \)).

5. **Establish Relationships**:
- From the perpendicular conditions, we can derive:
- \( a = f \) (from \( MN \))
- \( b = e \) (from \( PQ \))
- Additionally, since \( MN \) and \( PQ \) are midpoints, we can conclude:
- \( BC = AD \) (i.e., \( d = c \)).

6. **Summarize the Results**:
- From the analysis, we have the following equalities:
- \( AB = CD \)
- \( AC = BD \)
- \( BC = AD \)
- \( AN = BN \) (since \( M \) and \( N \) are midpoints, and angles \( \angle ABN = \angle ABM \) are equal).

### Conclusion:
Thus, the correct relationships are:
- \( AB = CD \)
- \( AC = BD \)
- \( BC = AD \)
- \( AN = BN \)