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.