Find the centroid of the triangle ABC whose vertices are A (9,2), B(1,10) and C(-7,-6). Find the coordinates of the middle points of its sides and hence find the centroid of the triangle formed by joining these middle points. Do the two triangles have the same centroid?

The vertices of a triangle are A (9,2), B(1,10) and C(-7,-6)

Here, x₁ = 9, x₂ = 1, x₃ = -7

and y₁ = 2, y₂ = 10, y₃ = -6

Let the coordinates of the centroid be(x,y)

So,

= (1,2)

Hence, the centroid of a triangle is (1, 2)

Now,

Let D, E and F are the midpoints of the sides BC, CA and AB respectively.

The coordinates of D are:

D = (-3, 2)

The coordinates of E are:

E = (1, -2)

The coordinates of F are:

F = (5, 6)

Now, we find the centroid of a triangle formed by joining these middle points D, E, and F as shown in figure

Let P be the trisection point of the median AD which is nearer to the opposite side BC

∴ P divides DA in the ratio 1:2 internally

= (1, 2)

Let Q be the trisection point of the median BE which is nearer to the opposite side CA

∴ Q divides EB in the ratio 1:2 internally

= (1, 2)

Let R be the trisection point of the median CF which is nearer to the opposite side AB

∴ R divides FC in the ratio 1:2 internally

= (1, 2)

Yes, the triangle has the same centroid, i.e. (1,2)