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In tetrahedron OA1A2A3, altitudes OP (from O to face A1A2A3) and A1Q (from A1 to face OA2A3) are drawn and they intersect at point H. Let OA1 = l1, A2A3 = l2, the shortest distance between lines OA1 and A2A3 be d, and the volume of the tetrahedron be V. Find the value of (d * l1 * l2) / V.
- 3
- 6
- 9
- 12
Correct answer: 6
Solution
The intersection of the two altitudes implies an orthocentric tetrahedron where opposite edges are mutually perpendicular (OA1 perp A2A3, etc.). The volume formula for skew perpendicular edges is V = (1/6)*d*l1*l2, giving d*l1*l2/V = 6.
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