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A tetrahedron ABCD has edges AB = CD = 12. These two edges are perpendicular to each other. E and F are the midpoints of AB and CD respectively. EF = 10 and EF is perpendicular to both AB and CD. Find the volume of tetrahedron ABCD.

1. 200
2. 240
3. 280
4. 320

Correct answer: 240

Solution

Set up coordinates with E at origin. AB is along x-axis: A = (-6,0,0), B = (6,0,0). CD is along y-axis (perpendicular to AB), with F at distance EF=10 along z-axis: F = (0,0,10), C = (0,-6,10), D = (0,6,10). Now compute volume of ABCD using V = (1/6)|det[AB, AC, AD]|. AB = B-A = (12,0,0). AC = C-A = (6,-6,10). AD = D-A = (6,6,10). Det = 12 * [(-6)(10) - (10)(6)] - 0 + 0 = 12*(-60-60) = 12*(-120) = -1440. Volume = (1/6)*1440 = 240.

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