Question

If the vertices of a triangle are $(a,b, - c),(c, - b,0)$ and $(b,0,c)$ , then find where the centroid of the triangle lies.
A. At origin
B. On x-axis
C. On y-axis
D. None of these

Answer 1

Hint: Add the x coordinates of the given points to obtain the x coordinate of the centroid of the triangle, similarly add y and z coordinates to obtain the y and z coordinates of the centroid. Then observe the obtained point for the required answer.

Formula Used:
The formula of the centroid of the triangle is,
$\left( {\dfrac{{{a_1} + {a_2} + {a_3}}}{3},\dfrac{{{b_1} + {b_2} + {b_3}}}{3},\dfrac{{{c_1} + {c_2} + {c_3}}}{3}} \right)$, where three vertices of the triangle are $\left( {{a_1},{b_1},{c_1}} \right),\left( {{a_2},{b_2},{c_2}} \right),\left( {{a_3},{b_3},{c_3}} \right)$.

Complete step by step solution:
The diagram of a triangle is,

Here, $A\left( {a,b, - c} \right),B(c, - b,0),C(b,0,c)$and G is the centroid of the triangle.
The formula to obtain the value of the centroid is $\left( {\dfrac{{{a_1} + {a_2} + {a_3}}}{3},\dfrac{{{b_1} + {b_2} + {b_3}}}{3},\dfrac{{{c_1} + {c_2} + {c_3}}}{3}} \right)$ , where three vertices of the triangle are $\left( {{a_1},{b_1},{c_1}} \right),\left( {{a_2},{b_2},{c_2}} \right),\left( {{a_3},{b_3},{c_3}} \right)$.
Now, substitute $A\left( {a,b, - c} \right),B(c, - b,0),C(b,0,c)$for $\left( {{a_1},{b_1},{c_1}} \right),\left( {{a_2},{b_2},{c_2}} \right),\left( {{a_3},{b_3},{c_3}} \right)$in the expression $\left( {\dfrac{{{a_1} + {a_2} + {a_3}}}{3},\dfrac{{{b_1} + {b_2} + {b_3}}}{3},\dfrac{{{c_1} + {c_2} + {c_3}}}{3}} \right)$to obtain the required value.
Therefore, the centroid is $\left( {\dfrac{{a + c + b}}{3},\dfrac{{b - b + 0}}{3},\dfrac{{ - c + 0 + c}}{3}} \right)$ , that is $\left( {\dfrac{{a + b + c}}{3},0,0} \right)$, this point lies on the x axis as the values of other coordinates are zero.

Option ‘B’ is correct

Note: Do not get confused that the point of the form $(a,0,0)$ lies in the x plane or in the x axis, so for this as the y and z coordinates are 0 these type of points always lie in the x axis.