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STATEMENT-1 : The centroid of a tetrahed...

STATEMENT-1 : The centroid of a tetrahedron with vertices (0, 0,0), (4, 0, 0), (0, -8, 0), (0, 0, 12)is (1, -2, 3).
and
STATEMENT-2 : The centroid of a triangle with vertices `(x_(1), y_(1), z_(1)), (x_(2), y_(2), z_(2)) and (x_(3), y_(3), z_(3))` is
`((x_(1)+x_(2)+x_(3))/3, (y_(1)+y_(2)+y_(3))/3, (z_(1)+z_(2)+z_(3))/3)`

A

Statement-1 is True, Statement-2 is true, Statement- is a correct explanation for Statement -1

B

Statement-1 is True, Statement-2 is true, Statement- is NOT a correct explanation for Statement -1

C

Statement-1 is True, Statement-2 is False

D

Statement-1 is False, Statement-2 is true

Text Solution

Verified by Experts

The correct Answer is:
B
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Explore conceptually related problems

Show that the coordinates off the centroid of the triangle with vertices A(x_(1),y_(1),z_(1)),B(x_(2),y_(2),z_(2)) and (x_(3),y_(3),z_(3)) are ((x_(1)+x_(2)+x_(3))/(3),(y_(1)+y_(2)+y_(3))/(3),(z_(1)+z_(2)+z_(3))/(3))

Find the co oridinate of the centroid of the tetrahedron whose vertices are (x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2)),(x_(3),y_(3),z_(3)) and (x_(4),y_(4),z_(4))

Knowledge Check

  • For x_(1), x_(2), y_(1), y_(2) in R if 0 lt x_(1)lt x_(2)lt y_(1) = y_(2) and z_(1) = x_(1) + i y_(1), z_(2) = x_(2)+ iy_(2) and z_(3) = (z_(1) + z_(2))//2, then z_(1) , z_(2) , z_(3) satisfy :

    A
    `abs(z_(1)) = abs( z_(2) ) = abs(z_(3))`
    B
    `abs(z_(1)) lt abs( z_(2) ) lt abs(z_(3))`
    C
    `abs(z_(1)) gt abs( z_(2) ) gt abs(z_(3))`
    D
    `abs(z_(1)) lt abs( z_(3) ) lt abs(z_(2))`

  • The value of |(2 y_(1)z_(1),y_(1) z_(2) + y_(2) z_(1),y_(1) z_(3) + y_(3) z_(1)),(y_(1) z_(2) y_(2) z_(1),2y_(2) z_(2),y_(2) z_(3) + y_(3) z_(2)),(y_(1) z_(3) + y_(3) z_(1),y_(2) z_(3) + y_(3) z_(2),2y_(3) z_(3))| , is

    A
    `y_(1) y_(2) y_(3) z_(1) z_(2) z_(3)`
    B
    `y_(1) + y_(2) + y_(3)`
    C
    `z_(1) + z_(2) + z_(3)`
    D
    0

  • A tetrahedron is three dimensional figure bounded by four non coplanar triangular plane. So a tetrahedron has points A,B,C,D as its vertices, which have coordinates (x_(1),y_(1),z_(1)) (x_(2), y_(2), z_(2)) , (x _(3), y_(3) , z_(3)) and (x _(4), y _(4), z _(4)) respectively in a rectangular three –dimensional space. Then the coordinates of its centroid are ((x_(1)+ x_(2) + x _(3) + x_(4))/(4) , (y _(1) + y _(2) + y_(3) + y _(4))/(4), (z_(1) + z_(2) + z_(3)+ z_(4))/(4)). The circumcentre of the tetrahedron is the centre of a sphere passing through its vertices. So, the circumcentre is a point equidistant from each of the vertices of tetrahedron. Let tetrahedron has three of its vertices represented by the points (0,0,0) ,(6,-5,-1) and (-4,1,3) and its centroid lies at the point (1,-2,5). Now answer the following questions The distance between the planes x + 2y- 3z -4 =0 and 2x + 4y - 6z=1 along the line x/1= (y)/(-3) = z/2 is :

    A
    `19/22`
    B
    `(7 sqrt7)/(22)`
    C
    `(7)/(22)`
    D
    `(7 sqrt14)/(22)`

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    Find the coordinates of the centroid of a triangle having vertices P(x_1,y_1,z_1),Q(x_2,y_2,z_2) and R(x_3,y_3,z_3)

    A=[[2,0,00,2,00,0,2]] and B=[[x_(1),y_(1),z_(1)x_(2),y_(2),z_(2)x_(3),y_(3),z_(3)]]

    Sow that x_(1)hat i+y_(1)hat j+z_(1)hat k,x_(2)hat i+y_(2)hat j+z_(2)hat k, and x_(3)hat i+y_(3)hat j+z_(3)hat k are non-coplanar if |x_(1)|>|y_(1)|+|z_(1)|,|y_(2)|>|x_(2)|+|z_(2)| and |z_(3)|>|x_(3)|+|y_(3)|

    If x_(1) = 3y_(1) + 2y_(2) -y_(3), " " y_(1)=z_(1) - z_(2) + z_(3) x_(2) = -y_(1) + 4y_(2) + 5y_(3), y_(2)= z_(2) + 3z_(3) x_( 3)= y_(1) -y_(2) + 3y_(3)," " y_(3) = 2z_(1) + z_(2) espress x_(1), x_(2), x_(3) in terms of z_(1) ,z_(2),z_(3) .

    A tetrahedron is three dimensional figure bounded by four non coplanar triangular plane. So a tetrahedron has points A,B,C,D as its vertices, which have coordinates (x_(1),y_(1),z_(1)) (x_(2), y_(2), z_(2)) , (x _(3), y_(3) , z_(3)) and (x _(4), y _(4), z _(4)) respectively in a rectangular three –dimensional space. Then the coordinates of its centroid are ((x_(1)+ x_(2) + x _(3) + x_(4))/(4) , (y _(1) + y _(2) + y_(3) + y _(4))/(4), (z_(1) + z_(2) + z_(3)+ z_(4))/(4)). The circumcentre of the tetrahedron is the centre of a sphere passing through its vertices. So, the circumcentre is a point equidistant from each of the vertices of tetrahedron. Let tetrahedron has three of its vertices represented by the points (0,0,0) ,(6,-5,-1) and (-4,1,3) and its centroid lies at the point (1,-2,5). Now answer the following questions The equation of the triangular plane of tetrahedron that contains the given vertices is :

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