If the mid-points of the sides $A B, B C, C A$ of a triangle are $(1,5,-1),(0,4,-2),(2,3,4)$ respectively, then the length of the median drawn from $C$ to $A B$ is

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Aicharya · Accepted Answer

$D(1,5,-1), E(0,4,-2), F(2,3,4)$ are mid-points Then, $ \begin{aligned} & \frac{x_1+x_2}{2}=1 \frac{y_1+y_2}{2}=5 \frac{z_1+z_2}{2}=-1 \\ & x_1+x_2=2, \quad y_1+y_2=10, \quad z_1+z_2=-2 \end{aligned} $ Similarly, $ \begin{array}{lll} x_2+x_3=0, & y_2+y_3=8, & z_2+z_3=-4 \\ x_1+x_3=4, & y_1+y_3=6, & z_1+z_3=8 \end{array} $ On adding, $2\left(x_1+x_2+x_3\right)=6$, $ \begin{aligned} & 2\left(y_1+y_2+y_3\right)=24, \quad 2\left(z_1+z_2+z_3\right)=2 \\ & x_1+x_2+x_3=3, \quad y_1+y_2+y_3=12 \\ & z_1+z_2+z_3=1 \\ & \Rightarrow \quad x_3=1, y_3=2, z_3=3 \\ & \end{aligned} $ length of median drawn from $C$ to $A B$ is $ \begin{aligned} C D & =\sqrt{(1-1)^2+(2-5)^2+(-1-3)^2} \\ & =\sqrt{0+9+16}=\sqrt{25}=5 \end{aligned} $

If the mid-points of the sides $A B, B C, C A$ of a triangle are $(1,5,-1),(0,4,-2),(2,3,4)$ respectively,…

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