Question

A(3, 1), B(y, 4) and C(1, x) are vertices of a triangle ABC. P, Q and R are mid - points of sides BC, CA and AB respectively. Show that the centroid of ΔPQR is the same as the centroid ΔABC.

Answer 1

Centroid of ΔABC = `((3 + y + 1)/3, (1+ 4 + x)/3) = ((4 + y)/3, (5 + x)/3)`

P, Q and R are the mid points of the sides BC, CA and AB.

By mid - point formula, we get 

`=> P = ((y + 1)/2, (4 + x)/2), Q = (4/2, (1 + x)/2) and R = ((3 + y)/2, 5/2)`

Centroid of a ΔPQR = `(((y + 1)/2 + 4/2 + (3 + y)/2)/3, ((4 + x)/2 - (1 + x)/2 + 5/2)/3)`

`= (((y + 1 + 4 + 3 + y)/2)/3, ((4 + x + 1 + x + 5)/2)/3)`

`= ((8 + 2y)/6, (10 + 2x)/6)`

`= ((4 + y)/3, (5 + x)/3)`               ........(ii)

From (i) and (ii), we get

Centroid of a ∆ABC = Centroid of a ∆PQR