The length of the altitude through the point D of tetrahedron where the vertices of the tetrahedron are A (2,3,1), B (4,1,-2), C (6,3,7) , D(-5,-4,8) , is

Step-by-step solution

The altitude length h through point D is found from the tetrahedron volume formula h = 3V Area _ ABC . Using D as the reference vertex, vectors are computed as: DA = (7, 7, -7) DB = (9, 5, -10) DC = (11, 7, -1) The scalar triple product gives: DA ( DB DC ) = (7, 7, -7) (65, -101, 8) = -308 Thus the volume is V = 1 6 |-308| = 154 3 . For the base triangle ABC , vectors from A are: AB = (2, -2, -3) AC = (4, 0, 6) The cross product magnitude is | AB AC | = |(-12, -24, 8)| = 28 , so the area is 1 2 28 = 14 . Substituti