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Centroid of a triangle or tetrahedron: find the unknown coordinates

This type of question, asked 5 times: 3 May 2023 S2, 15 May 2023 S1, 9 May 2024 S1, 12 May 2024 S1, 11 Apr 2026 S2. All 5 questions below, oldest first.

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  1. Question 1

    3 May 2023 S2

    143
    If A(−4,5,p), B(3,1,4)A(-4,5,p),\,B(3,1,4) and C(−2,0,q)C(-2,0,q) are the vertices of a triangle ABC and G(r,q,1)G(r,q,1) is its centroid, then the value of 2p+q−r2p+q-r is equal to

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  2. Question 2

    15 May 2023 S1

    143
    The centroid of tetrahedron with vertices P(5,-7,0), Q(a,5,3), R(4,-6,b) and S(6,c,2) is (4,-3,2), then the value of 2a+3b+c2a+3b+c is equal to

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  3. Question 3

    9 May 2024 S1

    139
    If two vertices of a triangle are A(3,1,4)A(3,1,4) and B(−4,5,−3)B(-4,5,-3) and the centroid of the triangle is G(−1,2,1)G(-1,2,1), then the third vertex C of the triangle is

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  4. Question 4

    12 May 2024 S1

    102
    The centroid of tetrahedron with vertices at A(−1,2,3)A(-1,2,3), B(3,−2,1)B(3,-2,1), C(2,1,3)C(2,1,3) and D(−1,−2,4)D(-1,-2,4) is

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  5. Question 5

    11 Apr 2026 S2

    94
    If the centroid of a tetrahedron OABCOABC is (1,2,−1)(1, 2, -1), where O is the origin, A(a,2,3),B(1,b,2),C(2,1,c)A(a, 2, 3), B(1, b, 2), C(2, 1, c) are the other vertices, then the distance of the point P(a,b,c)P(a, b, c) from the origin is...

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