The points A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) are the vertices of ΔABC.

(i) The median from A meets BC at D. Find the coordinates of the point D.

(ii) Find the coordinates of the point P on AD such that AP: PD = 2: 1.

(iii) Find the coordinates of points Q and R on medians BE and CF, respectively such that BQ: QE = 2:1 and CR: RF = 2: 1.

(iv) What are the coordinates of the centroid of the ΔABC?

Given,

The vertices of ΔABC = A, B and C

Coordinates of A, B and C;

A(x₁, y₁)

B(x₂, y₂)

C(x₃, y₃)

(i) As per given information D is the mid - point of BC and it bisect the line into two equal parts.

Coordinates of the mid - point of BC;

BC –

(ii) Let the coordinates of a point P be (x, y)

Given,

The point P(x, y), divide the line joining A(x₁, y₁) and D in the ratio 2:1

Then,

Coordinates of P =

By using internal section formula;

=

(iii) ∴ Let the coordinates of a point Q be (p, q)

Given,

The point Q (p, q),

Divide the line joining B(x₂, y₂) and E in the ratio 2:1,

Then,

Coordinates of Q =

=

Since, BE is the median of side CA, So BE divides AC in to two equal parts.

∴ mid - point of AC = Coordinate of E;

E =

So, the required coordinate of point Q;

Q =

Now,

Let the coordinates of a point E be (⍺, β)

Given,

Point R (⍺, β) divide the line joining C(x3, y3) and F in the ratio 2:1,

Then the coordinates of R;

=

=

Since, CF is the median of side AB.

So, CF divides AB in to two equal parts.

∴ mid - point of AB = Coordinate of F;

F =

So, the required coordinate of point R;

=

(iv) Coordinate of the centroid of the ΔABC;