Question

If $G\left( {3, - 5,r} \right)$ is centroid of triangle ABC where $A\left( {7, - 8,1} \right),B\left( {p,q,5} \right)$ and $C\left( {q + 1,5p,0} \right)$ are vertices of a triangle then values of $p,q,r$ are respectively.
A. $6,5,4$
B. $- 4,5,4$
C. $- 3,4,3$
D. $- 2,3,2$

Answer 1

We will use the formula of centroid, that is, if $\left( {{x_1},{y_1},{z_1}} \right),\left( {{x_2},{y_2},{z_2}} \right),\left( {{x_3},{y_3},{z_3}} \right)$ are the coordinates of vertices of a triangle, then the coordinates of centre are given by $\left( {\dfrac{{{x_1} + {x_2} + {x_3}}}{3},\dfrac{{{y_1} + {y_2} + {y_3}}}{3},\dfrac{{{z_1} + {z_2} + {z_3}}}{3}} \right)$. Substitute the given values of centroid and vertices of a triangle. Next, compare the coordinates to determine the value of $p, q, r$.

Complete step by step solution:
We are given that $G$ is the centroid of the triangle, whose vertices are $A,B$ and $C$.
A centroid is a point of intersection of three medians of a triangle.
Also, we know that if $\left( {{x_1},{y_1},{z_1}} \right),\left( {{x_2},{y_2},{z_2}} \right),\left( {{x_3},{y_3},{z_3}} \right)$ are the coordinates of vertices of a triangle, then the coordinates of centre are given by $\left( {\dfrac{{{x_1} + {x_2} + {x_3}}}{3},\dfrac{{{y_1} + {y_2} + {y_3}}}{3},\dfrac{{{z_1} + {z_2} + {z_3}}}{3}} \right)$
Then, from the given conditions, we will have,
$\left( {3, - 5,r} \right) = \left( {\dfrac{{7 + p + q + 1}}{3},\dfrac{{ - 8 + q + 5p}}{3},\dfrac{{1 + 5 + 0}}{3}} \right) \\\ \Rightarrow \left( {3, - 5,r} \right) = \left( {\dfrac{{8 + p + q}}{3},\dfrac{{ - 8 + q + 5p}}{3},2} \right)$
Now, we will compare the coordinates and form respective equations.
On comparing the $x$ coordinates, we will get,
$3 = \dfrac{{8 + p + q}}{3} \\\ \Rightarrow 9 = 8 + p + q$
$\Rightarrow p + q = 1$ eqn. (1)
On comparing the $y$ coordinates, we will get,
$\- 5 = \dfrac{{ - 8 + q + 5p}}{3} \\\ \Rightarrow - 15 = - 8 + q + 5p$
$\Rightarrow 5p + q = - 7$ eqn. (2)
On comparing the $z$ coordinates, we will get,
$r = 2$
We will solve equations (1) and (2) to determine the values of $p$ and $q$.
Now, subtract equations (1) and (2) to eliminate the value of $q$ and then find the value of $p$
$p + q - 5p - q = 1 - \left( { - 7} \right) \\\ \Rightarrow - 4p = 8 \\\ \Rightarrow p = - 2$
Substitute the value $p = - 2$ in equation(1) to find the value of $q$
$\- 2 + q = 1 \\\ \Rightarrow q = 3$
Therefore, the values of $p,q,r$ are $- 2, 3, 2$

Hence, option D is the correct option.

Note:
If two coordinates $\left( {{x_1},{y_1},{z_1}} \right)$ is equal to $\left( {{x_2},{y_2},{z_2}} \right)$, then ${x_1} = {x_2}$, ${y_1} = {y_2}$ and ${z_1} = {z_2}$. A centroid is a point of intersection of three medians of a triangle, where the median is a line from the vertex that divides the opposite side in two equal parts.