# Let A (4, 2), B(6, 5) and C (1, 4) be the vertices of Δ ABC(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 do you observe?[Note: The point which is common to all the three medians is called the centroidand this point divides each median in the ratio 2: 1](v) If and are the vertices of Δ ABC, find the coordinates of the centroid of the triangle

(i) Median AD of the triangle will divide the side BC in two equal parts

Therefore, D is the mid-point of side BC And according to midpoint formula, midpoints of (x1, y1), (x2, y2) is given by Coordinates of D = ( )

= ( )

(ii) Point P divides the side AD in a ratio 2:1

By section formula if (x, y) divides line joining points  A(x1, y1), B(x2, y2) in ratio m:n

then, Coordinates of P = ( )

= ( (iii) Median BE of the triangle will divide the side AC in two equal parts.

Therefore, E is the mid-point of side AC

Coordinates of E = ( , )

= ( )

Point Q divides the side BE in a ratio 2:1

By section formula if (x, y) divides line joining points  A(x1, y1), B(x2, y2) in ratio m:n

then, Coordinates of Q = ( )

= ( Median CF of the triangle will divide the side AB in two equal parts. Therefore, F is the mid-point of side AB

Coordinates of F = ( )

= (5, )

Point R divides the side CF in a ratio 2:1

By section formula if (x, y) divides line joining points  A(x1, y1), B(x2, y2) in ratio m:n

then, Coordinates of R = ( )

= ( )

(iv) It can be observed that the coordinates of point P, Q, R are the same.Therefore, all these are representing the same point on the plane i.e., the centroid of the triangle

(v) Consider a triangle, ΔABC, having its vertices as A(x1, y1), B(x2, y2), and C(x3,y3)

Median AD of the triangle will divide the side BC in two equal parts. Therefore, D is the mid-point of side BC

Coordinates of D = ( , )

Let the centroid of this triangle be O. Point O divides the side AD in a ratio 2:1

By section formula if (x, y) divides line joining points  A(x1, y1), B(x2, y2) in ratio m:n

then, Coordinates of O = ( )

Centroid of ABC = ( )

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