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Introduction To Three Dimensional Geometry

Short Answer Type

81. Find the co-ordinates of the centroid of triangle whose vertices are A (2, 3, -1), B (1, 3, -2) and C (3, 0, 3).

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82. The centroid of a triangle ABC is at the point (1, 1, 1). If the co-ordinates of A and B are (3, -5, 7) and (-1, 7, -6) respectively, find the co-ordinates of the point C

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83. If the origin is the centroid of triangle PQR, with vertices P(2a, 2, 6), Q(- 4, 3b, - 10) and R(8, 14, 2c). Find the values of a, b and c.

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84. Without using distance formula, show that A (-1, 2, 1), B (1, -2, 5) C (4, - 7, 8) and D (2, - 3, 4) are the vertices of a parallelogram ABCD.

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85. Three vertices of a parallelogram ABCD are A (3, -1, 2), B (1, 2, -4), and C (-1, 1, 2). Find the co-ordinates of the fourth vertex.

Three vertices of parllelogram ABCD are A (3, -1, 2), B (1, 2, -4), and C (-1, 1, 2).
Let the fourth vertex be D (x, y, z). ABCD is a parallelogram.
$rightwards double arrow$ Diagonals AC and BD bisect each other $rightwards double arrow$ Mid-point of AC and BD are coincident.
$rightwards double arrow$ $open parentheses fraction numerator 3 plus left parenthesis negative 1 right parenthesis over denominator 2 end fraction comma space fraction numerator negative 1 plus 1 over denominator 2 end fraction comma space fraction numerator 2 plus 2 over denominator 2 end fraction close parentheses space left right arrow space open parentheses fraction numerator 1 plus straight x over denominator 2 end fraction comma space fraction numerator 2 plus straight y over denominator 2 end fraction comma space fraction numerator negative 4 plus straight z over denominator 2 end fraction close parentheses space$
$rightwards double arrow$ $fraction numerator straight x plus 1 over denominator 2 end fraction space equals space 1 comma space fraction numerator straight y plus 2 over denominator 2 end fraction equals 0 comma space fraction numerator straight z minus 4 over denominator 2 end fraction equals 2$
$rightwards double arrow$ x = 1, y = -2, z = 8
Hence, vertex D is (1, -2, 8)
Using distance formula, we have
$AC space equals space square root of left parenthesis 3 plus 1 right parenthesis squared plus left parenthesis negative 1 minus 1 right parenthesis squared plus left parenthesis 2 minus 2 right parenthesis squared end root space equals space square root of 16 plus 4 end root space equals space square root of 20 space equals space 2 square root of 5$
$space space space space space space BD space equals space square root of left parenthesis 1 minus 1 right parenthesis squared plus left parenthesis 2 plus 2 right parenthesis squared plus left parenthesis negative 4 minus 8 right parenthesis squared end root space equals space square root of 0 plus 16 plus 144 end root space equals space square root of 160 space equals space 4 square root of 10$
$AC space not equal to space BD$
∴ ABCD is not a rectangle.

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Long Answer Type

86. The co-ordinates of the mid-points of sides BC, CA and AB of ∆ ABC are (2, 1, -1), (3, 2, 4) and (1, -2, 3) respectively. Find the co-ordinates of vertices A, B and C.

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Short Answer Type

87. Find the co-ordinates of a point P which divides the join of A (1, - 2, -1) and (1, 5, -8) internally in ratio 3 : 4.

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88. Find the co-ordinates of a point which divides the line joining points A (2, - 1, 3) and B (1, 4, 2) externally in ratio 2 : 3.

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89. Find the ratio in which the line joining points A (2, 4, - 3) and B (-3, 5, 2) is divided to XY-plane.

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90. Find the ratio in which the line joining points P (-3, 4, 5) and Q (5, 1, -1) divided by YZ-plane.

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