Q. 74.0( 77 Votes )

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

Answer :

(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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