Question

If the middle points of the sides of a triangle be (-2, 3), (4,-3) and (4, 5), then the centroid of the triangle is:

• A. (5/3,2)
• B. (5/6,1)
• C. (2,5/3)
• D. (1,5/6)

Answer 1

The correct option is C (2,5/3)

Let the vertices of the triangle be
$A(X_1,Y_1),B(X_2,Y_2)$ and $C(X_3,Y_3),$ then
$X_1+X_2=\mathrm{8...............}\left(1\right)Y_1+Y_2=\mathrm{10................}\left(2\right)X_2+X_3=-\mathrm{4...............}\left(3\right)Y_2+Y_3=\mathrm{6..................}\left(4\right)X_3+X_1=\mathrm{8...................}\left(5\right)Y_3+Y_1=-\mathrm{6.................}\left(6\right)$
On solving these equations, we get
$X_1=10,X_2=-2,X_3=-2,Y_1=-1,Y_2=11and{Y}_3=-5$
Hence, the centroid is (2, 5/3)

Examination method :
As we know that the centroid of the triangle ABC and that of the triangle formed by joining the middle points of the sides of triangle BC is same.
$\frac{4+4-2}{3},\frac{5-3+3}{3}=(2,\frac{5}{3})$