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Byju's Answer

Standard XII

Mathematics

Centroid of a Triangle

Three vertice...

Question

Three vertices of a tetrahedron are $(0, 0, 0), (6, - 5, - 1)$ and $(- 4, 1, 3)$ . If the centroid of the tetrahedron be $(1, - 2, 5)$ then the fourth vertex is

(2, - 4, 18)

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(1, - 4, 18)

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$(\frac{3}{2}, \frac{- 3}{2}, \frac{7}{4})$

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none of these

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Solution

The correct option is A $(2, - 4, 18)$
if the fourth vertex is $(x_{1}, y_{1}, z_{1})$
Then $0 + 6 - 4 + x_{1} = 0, ∴ x_{1} = 2$
similarily $0 - 5 + 1 + y_{1} = 0 ∴ y_{1} = - 4$
$0 - 1 + 3 + z_{1} = 20$
$∴ z_{1} = 18$
$∴$ the fourth vertex is $(2, - 4, 18)$

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Similar questions

Q. The equation of the triangular plane of tetrahedron that contains the given vertices

A (6, - 5, - 1), B (- 4, 1, 3)

C (2, - 4, 18)

A = (1, - 2, 3)

B =

(2, 1, 3),

C =

(4, 2, 1) and

G = (- 1, 3, 5)

is the centroid of the tetrahedron

A B C D

. Then the fourth coordinate is

Q. A tetrahedron is a three dimensional figure bounded by non coplanar triangular planes. So, a tetrahedron has four non-coplanar points as its vertices. Suppose a tetrehedron 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

[\frac{x_{1} + x_{2} + x_{3} + x_{4}}{4}, \frac{y_{1} + y_{2} + y_{3} + y_{4}}{4}, \frac{z_{1} + z_{2} + z_{3} + z_{4}}{4}]

.
Let a tetrahedron have 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 question. The coordinate of the fourth vertex of the tetrahedron is:

A tetrahedron is a three dimensional figure bounded by non coplanar triangular planes. So a tetrahedron has four non-coplanar points as its vertices. Suppose a tetrehedron has points $A, B, C, D$ as it 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 $[\frac{x_{1} + x_{2} + x_{3} + x_{4}}{4}, \frac{y_{1} + y_{2} + y_{3} + y_{4}}{4}, \frac{z_{1} + z_{2} + z_{3} + z_{4}}{4}]$ . Let $A$ 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 two questions. The equation of the traingular plane of tetrahedron that contains the given vertices

I

: Axes are translated to

(1, 0, - 1)

then the new coordinates of

(1, 2, - 3)

are

(0, 2, 2)

I I

: If

(3, 2, - 1), (4, 1, 1)

and

(6, 2, 5)

are three vertices of a tetrahedron and

(4, 2, 2)

is the centroid, then fourth vertex is

(3, 3, 3)

Which of the above statements is correct?

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Three vertices of a tetrahedron are (0,0,0),(6,−5,−1) and (−4,1,3). If the centroid of the tetrahedron be (1,−2,5) then the fourth vertex is

The correct option is A (2,−4,18)if the fourth vertex is (x1,y1,z1)Then 0+6−4+x1=0,∴x1=2similarily 0−5+1+y1=0∴y1=−40−1+3+z1=20∴z1=18∴ the fourth vertex is (2,−4,18)

Three vertices of a tetrahedron are $(0, 0, 0), (6, - 5, - 1)$ and $(- 4, 1, 3)$ . If the centroid of the tetrahedron be $(1, - 2, 5)$ then the fourth vertex is

The correct option is A $(2, - 4, 18)$
if the fourth vertex is $(x_{1}, y_{1}, z_{1})$
Then $0 + 6 - 4 + x_{1} = 0, ∴ x_{1} = 2$
similarily $0 - 5 + 1 + y_{1} = 0 ∴ y_{1} = - 4$
$0 - 1 + 3 + z_{1} = 20$
$∴ z_{1} = 18$
$∴$ the fourth vertex is $(2, - 4, 18)$