Question

The points $$(11,9),(2,1)$$ and $$(2,-1)$$ are the midpoints of the sides of the triangle.
Then the centroid is.
$$\begin{gathered} (A)(-5,-3)(B)(5,-3) \\ (C)(3,5)(D)(5,3) \end{gathered}$$

Answer 1

Hint:- Coordinates of midpoint of a line is $$\left({\displaystyle \frac{x_1+x_2}{2}},{\displaystyle \frac{y_1+y_2}{2}}\right)$$. 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 $$(a,b),(c,d)$$ 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 $$\left({\displaystyle \frac{a+c}{2}},{\displaystyle \frac{b+d}{2}}\right),\left({\displaystyle \frac{c+e}{2}},{\displaystyle \frac{d+f}{2}}\right)$$and $$\left({\displaystyle \frac{a+e}{2}},{\displaystyle \frac{b+f}{2}}\right).$$
As, we know that coordinates of centroid of the triangle are,
$$\Rightarrow$$Centroid of the triangle is $$\left({\displaystyle \frac{a+c+e}{3}},{\displaystyle \frac{b+d+f}{3}}\right)$$
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 $$\equiv \left({\displaystyle \frac{\left({\displaystyle \frac{a+c}{2}}\right)+\left({\displaystyle \frac{c+e}{2}}\right)+\left({\displaystyle \frac{a+e}{2}}\right)}{3}},{\displaystyle \frac{\left({\displaystyle \frac{b+d}{2}}\right)+\left({\displaystyle \frac{d+f}{2}}\right)+\left({\displaystyle \frac{b+f}{2}}\right)}{3}}\right)$$
So, putting the values of a, b and c in the above point denoted as centroid. We get,
$$\Rightarrow$$Centroid $$\equiv \left({\displaystyle \frac{11+2+2}{3}},{\displaystyle \frac{9+1-1}{3}}\right)\equiv \left(5,3\right)$$
$$\Rightarrow$$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.