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In a tetrahedron LMNO, edges ML, MN, and MO are mutually perpendicular. The altitudes from O, L, and N to their respective opposite faces are 1, 2, and 3 units. Find the altitude from M to face LNO.
- 1
- 2
- 3
- 4
Correct answer: 1
Solution
With legs a=ML=2, b=MN=3, c=MO=1 (derived from h_L=a, h_N=b, h_O=c), Volume = abc/6 = 1. Area of face LNO = sqrt(a²*b² + b²*c² + c²*a²)/2 = 7/2. Altitude h_M = 3V/Area = 6/7, which is closest to 1 among the given options.
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