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The equation for a median of a triangle is given by the midpoint formula:

[tex]\sf{\implies{ \text{Midpoint} = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right) }}[/tex]

The length of a median is then given by the distance formula:

[tex]\sf{\implies{ \text{Length} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} }}[/tex]

Let's calculate the length of median AM:

Step 1: Find the midpoint of BC

[tex]\sf{\implies{ M = \left(\frac{-6+8}{2}, \frac{5+7}{2}, \frac{3-7}{2}\right)

= (1, 6, -2) }}[/tex]

Step 2: Calculate the length of AM

[tex]\sf{\implies{ \text{Length of AM} = \sqrt{(2-1)^2 + (-3-6)^2 + (1+2)^2}}}[/tex]

= [tex]\sqrt{1 + 81 + 9}[/tex]

= [tex]\sf{\sqrt{91} }[/tex]

Length of median BM:

[tex]\sf{\implies{\text{Length of BM} = \sqrt{(-6+1)^2 + (5+6)^2 + (3+2)^2}}}[/tex]

= [tex]\sqrt{25 + 121 + 25}[/tex]

= [tex]\sqrt{171}[/tex]

Length of median CM:

[tex]\sf{\implies{ \text{Length of CM} = \sqrt{(8-1)^2 + (7+6)^2 + (-7+2)^2}}}[/tex]

= [tex]\sqrt{49 + 169 + 25}[/tex]

= [tex]\sqrt{243}[/tex]

Therefore, the lengths of the medians are [tex]\sqrt{91}[/tex], [tex]\sqrt{171}[/tex], and [tex]\sqrt{243}[/tex] units, respectively.

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