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# The points $(11,9),{\text{ }}(2,1)$ and $(2, - 1)$ are the midpoints of the sides of the triangle.Then the centroid is.$(A){\text{ }}( - 5, - 3){\text{ }}(B){\text{ }}(5, - 3) \\ (C){\text{ }}(3,5){\text{ }}(D){\text{ }}(5,3) \\$  Verified
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Hint:- Coordinates of midpoint of a line is $\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$. If coordinates of
the end points of the line are $({x_1},{y_1})$ and $({x_2},{y_2})$.

We are given with the coordinates of midpoints of the sides of the triangle.
Let the coordinates of the vertices of the triangle be,
$\Rightarrow$Vertices of the triangle are $(a,b),{\text{ }}(c,d)$ and $(e,f)$.
So, with the property of mid-point of the two given points.
We can write coordinates of mid-points of the sides of the triangle as,
$\Rightarrow$Midpoint of the sides will be $\left( {\dfrac{{a + c}}{2},\dfrac{{b + d}}{2}} \right),{\text{ }}\left( {\dfrac{{c + e}}{2},\dfrac{{d + f}}{2}} \right)$and $\left( {\dfrac{{a + e}}{2},\dfrac{{b + f}}{2}} \right).$
As, we know that coordinates of centroid of the triangle are,
$\Rightarrow$Centroid of the triangle is $\left( {\dfrac{{a + c + e}}{3},\dfrac{{b + d + f}}{3}} \right)$
And it can be easily seen that coordinates of the centroid of the triangle,
Can be easily obtained by adding the coordinates of the mid-points of its sides
and then dividing that by 3.
So, coordinates of centroid can be written as,
$\Rightarrow$Centroid $\equiv \left( {\dfrac{{\left( {\dfrac{{a + c}}{2}} \right) + \left( {\dfrac{{c + e}}{2}} \right) + \left( {\dfrac{{a + e}}{2}} \right)}}{3},\dfrac{{\left( {\dfrac{{b + d}}{2}} \right) + \left( {\dfrac{{d + f}}{2}} \right) + \left( {\dfrac{{b + f}}{2}} \right)}}{3}} \right)$
So, putting the values of a, b and c in the above point denoted as centroid. We get,
$\Rightarrow$Centroid $\equiv \left( {\dfrac{{11 + 2 + 2}}{3},\dfrac{{9 + 1 - 1}}{3}} \right) \equiv \left( {5,3} \right)$
$\Rightarrow$Hence, the coordinates of the centroid of the triangle will be $\left( {5,3} \right)$
$\Rightarrow$Hence, the correct option will be D.

Note:- Whenever we came up with this type of problem then first, we had to assume the
coordinates of vertices of triangle and then find mid-pints in terms of coordinates of
vertices. After that put coordinates of midpoints in terms of vertices of triangle in the formula
centroid triangle.