Question

Four vertices of a tetrahedron are $(0,0,0),(4,0,0),(0, - 8,0)$ and $(0,0,12)$ . It’s centroid has the coordinates
A. $\left( {\dfrac{4}{3},\dfrac{{ - 8}}{3},4} \right)$
B. $(2, - 4,6)$
C. $(1, - 2,3)$
D.None of these

Answer 1

Hint : As we know that a tetrahedron is a triangular period. It is also known as the triangular pyramid. We know that it is a polygon composed of four triangles, faces, six straight edges and four vertex corners. In this question we have to find the coordinates of the centroid. The formula is $\left( {\dfrac{{{x_1} + {x_2} + {x_3} + {x_4}}}{4},\dfrac{{{y_1} + {y_2} + {y_3} + {y_4}}}{4},\dfrac{{{z_1} + {z_2} + {z_3} + {z_4}}}{4}} \right)$ .

Complete step by step solution:
As per the question the four vertices of tetrahedron are $(0,0,0),(4,0,0),(0, - 8,0)$ and $(0,0,12)$ .
The formula we are using is
$\left( {\dfrac{{{x_1} + {x_2} + {x_3} + {x_4}}}{4},\dfrac{{{y_1} + {y_2} + {y_3} + {y_4}}}{4},\dfrac{{{z_1} + {z_2} + {z_3} + {z_4}}}{4}} \right)$ .
Here by comparing we have ${x_1} = 0,{x_2} = 4,{x_3} = 0,{x_4} = 0,{y_1} = 0,{y_2} = 0,{y_3} = - 8,{y_4} = 0,{z_1} = 0,{z_2} = 0,{z_3} = 0,{z_4} = 12$ .
Thus by substituting the vertices we get
$\left( {\dfrac{{0 + 4 + 0 + 0}}{4},\dfrac{{0 + 0 - 8 + 0}}{4},\dfrac{{0 + 0 + 0 + 12}}{4}} \right) = \left( {\dfrac{4}{4},\dfrac{{ - 8}}{4},\dfrac{{12}}{4}} \right)$
So the centroid of the coordinates is $(1, - 2,3)$ .
Hence the correct option is (c) $(1, - 2,3)$ .
So, the correct answer is “Option C”.

Note : Before solving this kind of questions we should have the idea of tetrahedron and their formulas. It is the simplest of the entire ordinary convex polygon and the only one that has fewer than $5$ faces. We should note that the tetrahedron is the $3D$ case of Euclideon simplex.