Question

Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0,4, 0) and (6, 0, 0).

Answer 1

Let AD, BE, and CF be the medians of the given triangle ABC. 
Since AD is the median, D is the mid-point of BC.
 ∴Coordinates of point D =(\(\frac{0+6}{2}\), \(\frac{4+0}{2}\),\(\frac{0+0}{2}\))= (3, 2, 0) 
AD = \(\sqrt{(0-3)^2+(0-2)^2+(6-0)^2}\) = \(\sqrt{9+4+36}\)= \(\sqrt{49}\) = 7 
Since BE is the median, E is the mid-point of AC. 
∴ Coordinates of point E = (\(\frac{0+6}{2}\), \(\frac{0+0}{2}\), \(\frac{6+0}{2}\)) = (3,0,3)
BE=\(\sqrt{(3-0)^2+(0-4)^2+(3-0)^2}\)=\(\sqrt{9+16+9}\) = \(\sqrt{34}\)
Since CF is the median, F is the mid-point of AB.
∴ Coordinates of point F =(\(\frac{0+0}{2}\), \(\frac{0+4}{2}\), \(\frac{6+0}{2}\)) =(0,2,3)
Length of CF = \(\sqrt{(6-0)+(0-2)+(0-3)}\)=\(\sqrt{36+4+9}\)=\(\sqrt{49}\)=7 
Thus, the lengths of the medians of ABC are 7,\(\sqrt{49}\), and 7.