1. Find the centroid of a triangle whose vertices are `A(4,-6),B(3,-2),C(5,2)`Solution:If `A(x_1,y_1),B(x_2,y_2)` and `C(x_3,y_3)` are the three vertices of the triangle ABC,
then the centroid of the triangle ABC is given by `G(x,y)=((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3)`
The given points are `A(4,-6),B(3,-2),C(5,2)`
`:. x_1=4,y_1=-6,x_2=3,y_2=-2,x_3=5,y_3=2`
Let `G(x,y)` be the centroid of the triangle ABC
`:. x=(x_1+x_2+x_3)/3`
`=(4+3+5)/3`
`=12/3`
`=4`
`:. y=(y_1+y_2+y_3)/3`
`=(-6-2+2)/3`
`=-6/3`
`=-2`
`:.` Centroid of triangle ABC is `G(4,-2)`
2. Find the centroid of a triangle whose vertices are `A(3,-5),B(-7,4),C(10,-2)`Solution:If `A(x_1,y_1),B(x_2,y_2)` and `C(x_3,y_3)` are the three vertices of the triangle ABC,
then the centroid of the triangle ABC is given by `G(x,y)=((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3)`
The given points are `A(3,-5),B(-7,4),C(10,-2)`
`:. x_1=3,y_1=-5,x_2=-7,y_2=4,x_3=10,y_3=-2`
Let `G(x,y)` be the centroid of the triangle ABC
`:. x=(x_1+x_2+x_3)/3`
`=(3-7+10)/3`
`=6/3`
`=2`
`:. y=(y_1+y_2+y_3)/3`
`=(-5+4-2)/3`
`=-3/3`
`=-1`
`:.` Centroid of triangle ABC is `G(2,-1)`
3. Find the centroid of a triangle whose vertices are `A(4,-8),B(-9,7),C(8,13)`Solution:If `A(x_1,y_1),B(x_2,y_2)` and `C(x_3,y_3)` are the three vertices of the triangle ABC,
then the centroid of the triangle ABC is given by `G(x,y)=((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3)`
The given points are `A(4,-8),B(-9,7),C(8,13)`
`:. x_1=4,y_1=-8,x_2=-9,y_2=7,x_3=8,y_3=13`
Let `G(x,y)` be the centroid of the triangle ABC
`:. x=(x_1+x_2+x_3)/3`
`=(4-9+8)/3`
`=3/3`
`=1`
`:. y=(y_1+y_2+y_3)/3`
`=(-8+7+13)/3`
`=12/3`
`=4`
`:.` Centroid of triangle ABC is `G(1,4)`
4. Find the centroid of a triangle whose vertices are `A(3,-7),B(-8,6),C(5,10)`Solution:If `A(x_1,y_1),B(x_2,y_2)` and `C(x_3,y_3)` are the three vertices of the triangle ABC,
then the centroid of the triangle ABC is given by `G(x,y)=((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3)`
The given points are `A(3,-7),B(-8,6),C(5,10)`
`:. x_1=3,y_1=-7,x_2=-8,y_2=6,x_3=5,y_3=10`
Let `G(x,y)` be the centroid of the triangle ABC
`:. x=(x_1+x_2+x_3)/3`
`=(3-8+5)/3`
`=0/3`
`=0`
`:. y=(y_1+y_2+y_3)/3`
`=(-7+6+10)/3`
`=9/3`
`=3`
`:.` Centroid of triangle ABC is `G(0,3)`
5. Find the centroid of a triangle whose vertices are `A(2,4),B(6,4),C(2,0)`Solution:If `A(x_1,y_1),B(x_2,y_2)` and `C(x_3,y_3)` are the three vertices of the triangle ABC,
then the centroid of the triangle ABC is given by `G(x,y)=((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3)`
The given points are `A(2,4),B(6,4),C(2,0)`
`:. x_1=2,y_1=4,x_2=6,y_2=4,x_3=2,y_3=0`
Let `G(x,y)` be the centroid of the triangle ABC
`:. x=(x_1+x_2+x_3)/3`
`=(2+6+2)/3`
`=10/3`
`:. y=(y_1+y_2+y_3)/3`
`=(4+4)/3`
`=8/3`
`:.` Centroid of triangle ABC is `G(10/3,8/3)`
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