Consider the tetrahedron with the vertices $\mathrm{A}(3,2,4)$, $\mathrm{B}\left(x_1, y_1, 0\right),…
Consider the tetrahedron with the vertices $\mathrm{A}(3,2,4)$, $\mathrm{B}\left(x_1, y_1, 0\right), \mathrm{C}\left(x_2, y_2, 0\right), \mathrm{D}\left(x_3, y_3, 0\right)$. If the triangle BCD is formed by the lines $y=x, x+y=6$ and $\mathrm{y}=1$, then the centroid of the tetrahedron is
$\left(\frac{9}{4}, \frac{7}{4}, 1\right)$
$\left(\frac{11}{4}, \frac{5}{4}, 1\right)$
$\left(3, \frac{7}{4}, 1\right)$
$(3,2,1)$
Solution
$\mathrm{A}(3,2,4), \mathrm{B}\left(x_1, y_1, 0\right), \mathrm{C}\left(x_2, y_2, 0\right), \mathrm{D}\left(x_3, y_3, 0\right)$
$\triangle \mathrm{BCD}$ is formed by lines $y=x, x+y=6, y=1$
$\therefore$ Vertices of BCD are D $(1,1), \mathrm{C}(3,3), \mathrm{D}(5,1)$
Let $\vec{l}, \vec{m}, \vec{n}, \vec{p}$ be position vectors of vertices $\mathrm{A} ; \mathrm{B}, \mathrm{C}$, and D
$\begin{aligned}
& \therefore \vec{l}=3 \hat{i}+2 \hat{j}+4 \hat{k}, \vec{m}=\hat{i}+\hat{j} \\
& \vec{h}=3 \hat{i}+3 \hat{j}, \vec{p}=5 \hat{i}+\hat{j}
\end{aligned}$
$\therefore$ position vector of centroid $\vec{g}=\frac{\vec{l}+\vec{m}+\vec{n}+\vec{p}}{4}$
$\Rightarrow \vec{g}=\frac{12 \hat{i}+7 \hat{j}+4 \hat{k}}{4}$
$\therefore$ Co-ordinates of centroid is $\left(3, \frac{7}{4}, 1\right)$