Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

The equation of the plane is $2x+3y+4z=1$.

To find the point $A$ on the X-axis, substitute $y=0$ and $z=0$:

$2x=1\Rightarrow x=\frac{1}{2}$

Thus, $A\equiv \left(\frac{1}{2},0,0\right)$.

To find the point $B$ on the Y-axis, substitute $x=0$ and $z=0$:

$3y=1\Rightarrow y=\frac{1}{3}$

Thus, $B\equiv \left(0,\frac{1}{3},0\right)$.

To find the point $C$ on the Z-axis, substitute $x=0$ and $y=0$:

$4z=1\Rightarrow z=\frac{1}{4}$

Thus, $C\equiv \left(0,0,\frac{1}{4}\right)$.

The centroid of $△ABC$ is given by $\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3}\right)$.

Substituting the coordinates of $A$, $B$, and $C$:

Centroid $=\left(\frac{\frac{1}{2}+0+0}{3},\frac{0+\frac{1}{3}+0}{3},\frac{0+0+\frac{1}{4}}{3}\right)$

Centroid $=\left(\frac{1}{6},\frac{1}{9},\frac{1}{12}\right)$

Answer: $\left(\frac{1}{6},\frac{1}{9},\frac{1}{12}\right)$