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 CR : RF = 2 : 1.


(iv) What do you observe?

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


Here given vertices are A (4, 2), B (6, 5) and C (1, 4).



By midpoint formula.


x = , y =


For midpoint D of side BC,


x = , y =


x = , y =


Hence, the coordinates of D are ( , )


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



By section formula,


x = , y =


For point P on AD, where m = 2 and n = 1


x = , y =


x = and y =


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


By midpoint formula.


x = , y =


For midpoint E of side AC,


x = , y =


x = , y =


Hence, the coordinates of E are ( , 3)


For midpoint F of side AB,


x = , y =


x = , y =


Hence, the coordinates of F are ( , )


By section formula,


x = , y =


For point Q on BE, where m = 2 and n = 1


x = , y =


x = and y =


For point R on CF, where m = 2 and n = 1


x = , y =


x = and y =


(iv) What do you observe?


We observe that the point P,Q and R coincides with the centroid.


This also shows that centroid divides the median in the ratio 2:1


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