3D Geometry
Tetrahedron properties
Grade 12
Question:
<p>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:</p>
<p>(a) AB = CD</p>
<p>(b) BC = DA</p>
<p>(c) AC = BD</p>
<p>(d) AN = BN</p>
Step-by-Step Solution
Key Concept: Perpendicularity conditions on midlines of a tetrahedron lead to equality of opposite edges.
Based on the perpendicularity conditions given, all four statements are correct. When MN ⊥ AB and MN ⊥ CD, we get AB = CD. When PQ ⊥ AC and PQ ⊥ BD, we get AC = BD. These conditions together imply BC = DA and AN = BN.
Correct Answer: a, b, c, d