Question

In a three-dimensional coordinate system P, Q and R are images of a point A (a, b, c) in the xy, yz and zx planes respectively. If G is the centroid of triangle PQR then area of triangle AOG is (O is the origin)

A.0 \\\ B.{a^2} + {b^2} + {c^2} \\\ C.\dfrac{2}{3}({a^2} + {b^2} + {c^2}) \\\ D.2({a^2} + {b^2} + {c^2}) \\\ } $$

Answer 1

At first, read the question carefully & note down the given data in the question & also what has to be found as an answer. Then, we have to find P and we will find this by finding the image in $xy$-plane. Then $z$coordinate will be replaced by $( - z)$coordinate then we have p $(a,b,c)$. In this way we can find the coordinate of point Q and R.

Complete step-by-step answer:
First, we have to find the coordinate of P. Since P is in $xy$-plane. Therefore, $z$ can be replaced by $( - z)$Therefore, Coordinate of P is $(a,b, - c)$
Similarly, for the Q coordinate
$y$ will be replaced by $( - y)$
Therefore, coordinate of Q is (a, - b,c)$$$$$ Similarly, for the R coordinate x$will be replaced by$( - x)$Therefore, coordinate of R is$( - a,b,c)$To find the centroid, we know the formula i.e.,$(\dfrac{{{x_1} + {x_2} + {x_3}}}{3},\dfrac{{{y_1} + {y_2} + {y_3}}}{3},\dfrac{{{z_1} + {z_2} + {z_3}}}{3})$Therefore, coordinate of centroid$(G)$, where$G = $Centroid${
= (\dfrac{{a - a + a}}{3},\dfrac{{b - b + b}}{3},\dfrac{{c + c - c}}{3}) \\
= (\dfrac{a}{3},\dfrac{b}{3},\dfrac{c}{3}) \\
} $We know that coordinate of origin in Three Dimension is$(0,0,0)$Let$O = (0,0,0)$[Where$O$is origin] Therefore, we have$A(a,b,c),O(0,0,0),G(\dfrac{a}{3},\dfrac{b}{3},\dfrac{c}{3})$To find the direction cosine, we can use the above expression. Therefore, direction cosine of$GO = (\dfrac{a}{3},\dfrac{b}{3},\dfrac{c}{3})$Therefore, direction cosine of$AO = (a,b,c)$ and direction cosine of $AG = (\dfrac{{2a}}{3},\dfrac{{2b}}{3},\dfrac{{2c}}{3})$
Now, From the above direction cosine, it has concluded that all the points lie on the same straight line.
Therefore, it is collinear.

Hence, the area of the triangle $AOG$ is $0$.

Note: The area of any collinear points is zero. Concept of straight line & its formulations should be known. It will be absolutely zero because it does not have any area. By using this method, we can easily reach out to the final answer.