A (4, 2), B (6, 5) and C (1, 4) are the vertices of △ABC.
(i) The median from A meets BC in D. Find the coordinates of the point D.
(ii) Find the coordinates of point P on AD such that AP : PD = 2:1.
(iii) Find the coordinates of the points Q and R on medians BE and CF respectively such that BQ : QE = 2 :1 and RF = 2 :1.
(iv) What do you observe?
A (4, 2), B (6, 5) and C (1, 4) are the vertices of △ABC.
(i) The median from A meets BC in D. Find the coordinates of the point D.
(ii) Find the coordinates of point P on AD such that AP : PD = 2:1.
(iii) Find the coordinates of the points Q and R on medians BE and CF respectively such that BQ : QE = 2 :1 and RF = 2 :1.
(iv) What do you observe?
(i) Median AD of the triangle will divide the side BC in two equal parts.
Therefore, D is the mid-point of side BC.
(ii) Point P divides the side AD in a ratio 2:1.
(iii) Median BE of the triangle will divide the side AC in two equal parts.
Therefore, E is the mid-point of side AC.
Point Q divides the side BE in a ratio 2:1.
Median CF of the triangle will divide the side AB in two equal parts. Therefore, F is the mid-point of side AB.
Point R divides the side CF in a ratio 2:1.
(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.
(i) Median AD of the triangle will divide the side BC in two equal parts.
Therefore, D is the mid-point of side BC.
(ii) Point P divides the side AD in a ratio 2:1.
(iii) Median BE of the triangle will divide the side AC in two equal parts.
Therefore, E is the mid-point of side AC.
Point Q divides the side BE in a ratio 2:1.
Median CF of the triangle will divide the side AB in two equal parts. Therefore, F is the mid-point of side AB.
Point R divides the side CF in a ratio 2:1.
(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.
