# Let a (4, 2), B (6, 5) and C (1, 4) Be the Vertices δAbc. (I) the Median from a Meets Bc at D. Find the Coordinates of Point D. (Ii) Find the Coordinates of the Point P on Ad Such that Ap: Pd = 2:1 - Mathematics

Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ΔABC.

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

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

(iii) Find the coordinates of point 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?

(v) If A(x1y1), B(x2y2), and C(x3y3) are the vertices of ΔABC, find the coordinates of the centroid of the triangle.

A (4, 2), B(6, 5) and C (1, 4) are the vertices of ΔABC.

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

(ii) Find the coordinates of point P and 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?

#### Solution 1 (i) Median AD of the triangle will divide the side BC in two equal parts.

Therefore, D is the mid-point of side BC

Coordinate of D =  ((6+1)/2, (5+4)/2) = (7/2, 9/2)

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

Coordinate of P =((2xx7/2+1xx4)/(2+1), (2xx9/2+1xx2)/(2+1))= (11/3, 11/3)

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

Therefore, E is the mid-point of side AC.

Coordinate of E = ((4+1)/2,(2+4)/2) = (5/2, 3)

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

Coordinate of Q =((2xx5/2+1xx6)/(2+1), (2xx3+1xx5)/(2+1)) = (11/3,11/3)

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

Coordinate of F =((4+6)/2, (2+5)/2) = (5, 7/2)

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

Coordinate of R =((2xx5+1xx1)/(2+1),(2xx7/2+1xx4)/(2+1)) = (11/3, 11/3)

(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(x1y1), B(x2y2), 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.

Coordinate of D =((x_2+x_3)/2,(y_2+y_3)/2)

Let the centroid of this triangle be O.

Point O divides the side AD in a ratio 2:1.

Coordinate of O = (2xx(x_2+x_3)/2+1xxx_1)/(2+1), (2xx(y_2+y_3)/2+1xxy_1)/(2+1))

= ((x_1+x_2+x_3)/3, (y_1+y_2+y_3)/3)

#### Solution 2

We have triangle ΔABC   in which the co-ordinates of the vertices are A (4, 2); B (6, 5) and C (1, 4)

(i)It is given that median from vertex A meets BC at D. So, D is the mid-point of side BC.

In general to find the mid-point p ( x, y)  of two points A( x1 , y1) and B (x2 , y2)  we use section formula as,

p ( x, y)  = ((x_1 + x_2 ) / 2 ,(y_1 + y_2) /2)

Therefore mid-point D of side BC can be written as,

D ( x, y) = ((6+1)/2 , (5+4)/2)

Now equate the individual terms to get,

 x = 7/2

y = 9/2

So co-ordinates of D is  (7/2 , 9/2)

(ii)We have to find the co-ordinates of a point P which divides AD in the ratio 2: 1 internally.

Now according to the section formula if any point P divides a line segment joining  A( x1 , y1) and B (x2 , y2) in the ratio m: n internally than,\

p ( x, y) = ((nx_1 + mx_2)/(m + n ) , ( ny_1 + my_2)/(m + n ))

P divides AD in the ratio 2: 1. So,

P(x , y) = ((2(7/2) + 4(1))/(1 + 2) , (2(9/2) + 1(2) ) /(1+2))

 = (11/3 , 11/3)

(iii)We need to find the mid-point of sides AB and AC. Let the mid-points be F and E for the sides AB and AC respectively.

Therefore mid-point F of side AB can be written as,

F ( x , y) = ((6+4)/2 , (5+ 2)/2)

So co-ordinates of F is   (5 , 7/2)

Similarly mid-point E of side AC can be written as,

E ( x , y) = ((1+4)/2 , (4 + 2)/2)

So co-ordinates of E is  (5/2 , 3)

Q divides BE in the ratio 2: 1. So,

 Q ( x , y) = (2(5/2) +6(1) )/(1+2), (2(3)+1 (5)) / (1+2)= (11/3, 11/3)

Similarly, R divides CF in the ratio 2: 1. So,

 R ( x , y) = (2(5) +1(1) )/(1+2), (2(7/2)+1 (4)) / (1+2)= (11/3, 11/3)

(iv)We observe that that the point P, Q and R coincides with the centroid. This also shows that centroid divides the median in the ratio 2: 1.

Concept: Area of a Triangle
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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 6 Co-Ordinate Geometry
Exercise 6.3 | Q 54 | Page 31
NCERT Class 10 Maths
Chapter 7 Coordinate Geometry
Exercise 7.4 | Q 7 | Page 171