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 step by step, we will analyze the tetrahedron ABCD and the midpoints M, N, P, and Q of its edges.

### Step 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.

### Step 2: Draw the tetrahedron
- Draw tetrahedron ABCD with vertices A, B, C, and D.
- Mark the midpoints M, N, P, and Q on the respective edges.

### Step 3: Analyze the given conditions
- We know that MN is perpendicular to both AB and CD.
- We also know that PQ is perpendicular to both AC and BD.

### Step 4: Establish properties of the tetrahedron
- Since MN is perpendicular to both AB and CD, it indicates a specific relationship between these edges, suggesting that the tetrahedron may be equilateral.
- Similarly, since PQ is perpendicular to both AC and BD, it reinforces the idea that the tetrahedron has equal edge lengths.

### Step 5: Conclude about the tetrahedron
- From the properties of the tetrahedron, we can conclude that:
1. AB = CD (since MN is perpendicular to both).
2. AC = BD (since PQ is perpendicular to both).

### Step 6: Analyze angles and sides
- In an equilateral tetrahedron, all angles are equal (60 degrees).
- Since angles ABN and BAN are equal, we can conclude that the sides opposite these angles are also equal, leading to:
- BN = AN.

### Step 7: Final relationships
- Since we have established that:
- AB = CD,
- AC = BD,
- BN = AN,
- and by symmetry and the properties of an equilateral tetrahedron, we can also conclude that:
- BC = DA.

### Conclusion
From the analysis, we conclude that all the relationships hold true. Therefore, all the options provided in the question are correct.