$A(5, 4, 6), B=(1, -1, 3)$ and $C(4, 3, 2)$ form $\Delta$ABC. If the internal bisector of angle A meets BC in D, then the length of $\overline{AD}$ is?

First, find the lengths of the sides AB and AC using the distance formula. AB = sqrt((5 - 1)^2 + (4 - (-1))^2 + (6 - 3)^2) = sqrt(16 + 25 + 9) = sqrt(50), and AC = sqrt((5 - 4)^2 + (4 - 3)^2 + (6 - 2)^2) = sqrt(1 + 1 + 16) = sqrt(18). By the Angle Bisector Theorem, the point D divides BC in the ratio of AB to AC, which simplifies to sqrt(50) : sqrt(18) = 5 : 3. Using the section formula, the coordinates of D are ((5*4 + 3*1)/8, (5*3 + 3*(-1))/8, (5*2 + 3*3)/8) = (23/8, 12/8, 19/8). The length of AD is calculated as sqrt((5 - 23/8)^2 + (4 - 12/8)^2 + (6 - 19/8)^2) = sqrt((17/8)^2 + (20/8)^2 + (29/8)^2) = sqrt(289 + 400 + 841)/8 = sqrt(1530)/8 = 3*sqrt(170)/8.

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