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# If (−2, 3), (4, −3) and (4, 5) Are the Mid-points of the Sides of a Triangle, Find the Coordinates of Its Centroid. - CBSE Class 10 - Mathematics

ConceptConcepts of Coordinate Geometry

#### Question

If (−2, 3), (4, −3) and (4, 5) are the mid-points of the sides of a triangle, find the coordinates of its centroid.

#### Solution

Let ΔABC be ant triangle such that P (−2, 3); Q (4,−3) and R (4, 5) are the mid-points of the sides AB, BC, CA respectively.

We have to find the co-ordinates of the centroid of the triangle.

Let the vertices of the triangle beA(x_1,y_1);B(x_2,y_2);C(x_3,y_3)

In general to find the mid-point p(x,y)  of two pointsA(x_1,y_1)andB(x_2,y_2) we use section formula as,

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

So, co-ordinates of P,

(-2,3)=((x_1+x_2)/2,(y_1+y_2)/2)

Equate the x component on both the sides to get,

x_1+x_2=-4 .........(1)

Similarly,

y_1+y_2=6 ..........(2)

Similary, co-ordinates of Q

(4,-3)=((x_3+x_2)/2,(y_3+y_2)/2)

Equate the x component on both the sides to get,

x_3+x_2=8.........(3)

Similarly,

y_3+y_2=-6 ..........(4)

Equate the x componet on both the sides to get,

x_3+x_1=8..........(5)

Similarly,

y_3+y_1=10..........(6)

Add equation (1) (3) and (5) to get,

2(x_1+x_2+x_3)=12

x_1+x_2+x_3 =6

Similarly, add equation (2) (4) and (6) to get,

2(y_1+y_2+y_3)=10

y_1+y_2+y_3=5

We know that the co-ordinates of the centroid G of a triangle whose vertices are

(x_1,y_1), (x_2,y_2),(x_3,y_3) is

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

So, centroid Gof a triangle triangle ABC is ,

G(2,5/3)

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Solution If (−2, 3), (4, −3) and (4, 5) Are the Mid-points of the Sides of a Triangle, Find the Coordinates of Its Centroid. Concept: Concepts of Coordinate Geometry.
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