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 (x

_{1}, y

_{1}), (x

_{2}, y

_{2}) 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(*x*,

_{1}*y*), B(

_{1}*x*,

_{2}*y*) in ratio m:n

_{2}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(*x*,

_{1}*y*), B(

_{1}*x*,

_{2}*y*) in ratio m:n

_{2}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(*x*,

_{1}*y*), B(

_{1}*x*,

_{2}*y*) in ratio m:n

_{2}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(*x _{1}*,

*y*), B(

_{1}*x*,

_{2}*y*), and C(

_{2}*x*,

_{3}*y*)

_{3}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(*x _{1}*,

*y*), B(

_{1}*x*,

_{2}*y*) in ratio m:n

_{2}then,

Coordinates of O = ()

Centroid of ABC = ()

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