<p>ABCD is a regular tetrahedron, A is the origin and B lies on x-axis. ABC lies in the xy-plane and <span class="latex">\(|\vec{AB}| = 2\)</span>. Under these conditions, the number of possible tetrahedrons is:</p>
Step-by-Step Solution
Key Concept: A regular tetrahedron has fixed edge length and fixed angles. Once A is at origin, B on x-axis, and ABC in xy-plane are fixed, point D can be positioned in multiple ways above or below the xy-plane, and C has discrete choices due to geometric constraints.
Step 1: Set up coordinates. A is at origin (0,0,0), B lies on x-axis with |AB|=2, so B = (2,0,0). Step 2: Triangle ABC lies in xy-plane. Since ABCD is regular with edge length 2, point C must satisfy: |AC|=2, |BC|=2, and C in xy-plane. This means C lies on the intersection of circles: center A radius 2 and center B radius 2 in the xy-plane. Step 3: Find C's position. The two circles intersect at points where C = (1, √3, 0) or C = (1, -√3, 0). This gives 2 choices for C . Step 4: For each choice of C, find D. Point D must satisfy |AD|=2, |BD|=2, |CD|=2. The locus of points equidistant from A, B, C is a line perpendicular to plane ABC passing through the circumcenter of triangle ABC. Step 5: The circumcenter of equilateral triangle ABC (with side 2) is at (1, √3/3, 0) or (1, -√3/3, 0) respectively. Point D lies on the perpendicular to xy-plane through this center. Step 6: For a regular tetrahedron with edge 2, the height from D to plane ABC is h = √(8/3). Therefore D has z-coordinate = ±√(8/3), giving 2 positions for D (above and below the plane). Step 7: Total number of configurations = 2 (choices for C) × 2 (orientations for each C) × 2 (D above or below plane) = 8. Step 8: Account for all valid geometric arrangements. Each distinct choice of (C, D) pair gives a valid regular tetrahedron satisfying all constraints. ∴ Answer: 8
Correct Answer: 8