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Introduction To Three Dimensional Geometry

Short Answer Type

71. Find the co-ordinates of the point which divides the line segment formed by joining the points (- 2, 3, 5) and (1, -4, 6) in the ratio of 2 : 3 externally.

117 Views

72. Given that P (3, 2, -4), Q (5, 4, -6) and R (9, 8, -10) are collinear. Find the ratio in which Q divides PR.

176 Views

73. Using section formula, prove that the three points (-4, 6, 10), (2, 4, 6) and (14, 0, - 2) are collinear. Also find the ratio in which point B divides the join of A and C.

719 Views

74. Find the co-ordinates of the points of trisection of the line segment joining the points P (3, 2, - 1) and Q (1, 2, 5).

125 Views

75.

Find the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8).

400 Views

76.

Find the ratio in which the join of P (2, 1, -1) and Q (3, 2, 4) is divided by XY-plane.

120 Views

77. A point R with x-coordinate 4 lies on the line joining the points P (2, - 3, 4) and Q (8, 0, 10). Find the co-ordinates of point R.

117 Views

Long Answer Type

78. Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and C (6, 0, 0).

straight A space left parenthesis 0 comma space 0 comma space 6 right parenthesis space left right arrow space left parenthesis straight x subscript 1 comma space straight y subscript 1 comma space straight z subscript 1 right parenthesis straight B space left parenthesis 0 comma space 4 comma space 0 right parenthesis space left right arrow space left parenthesis straight x subscript 2 comma space straight y subscript 2 comma space straight z subscript 2 right parenthesis straight C space left parenthesis 6 comma space 0 comma space 0 right parenthesis space left right arrow space left parenthesis straight x subscript 3 comma space straight y subscript 3 comma space straight z subscript 3 right parenthesis

D is mid-point of BC

rightwards double arrow

D is

open parentheses fraction numerator straight x subscript 2 plus straight x subscript 3 over denominator 2 end fraction comma space fraction numerator straight y subscript 2 plus straight y subscript 3 over denominator 2 end fraction comma space fraction numerator straight z subscript 2 plus straight z subscript 3 over denominator 2 end fraction close parentheses space left right arrow space open parentheses fraction numerator 0 plus 6 over denominator 2 end fraction comma space fraction numerator 4 plus 0 over denominator 2 end fraction comma space fraction numerator 0 plus 0 over denominator 2 end fraction close parentheses

= (3, 2, 0)

Using distance formula, we have

AD equals square root of left parenthesis 0 minus 3 right parenthesis squared plus left parenthesis 0 minus 2 right parenthesis squared plus left parenthesis 6 minus 0 right parenthesis squared end root space equals space square root of 9 plus 4 plus 36 end root space equals space square root of 49 space equals space 7

E is the mid-point of CA.

rightwards double arrow

straight E space equals space open parentheses fraction numerator straight x subscript 3 plus straight x subscript 1 over denominator 2 end fraction comma space fraction numerator straight y subscript 3 plus straight y subscript 1 over denominator 2 end fraction comma space fraction numerator straight z subscript 3 plus straight z subscript 1 over denominator 2 end fraction close parentheses space left right arrow space open parentheses fraction numerator 6 plus 0 over denominator 2 end fraction comma space fraction numerator 0 plus 0 over denominator 2 end fraction comma space fraction numerator 0 plus 6 over denominator 2 end fraction close parentheses space equals space left parenthesis 3 comma space 0 comma space 3 right parenthesis

∴

BE space equals space square root of left parenthesis 0 minus 3 right parenthesis squared plus left parenthesis 4 minus 0 right parenthesis squared plus left parenthesis 0 minus 3 right parenthesis squared end root space equals space square root of 9 plus 16 plus 9 end root space equals space square root of 34

F is the mid-point of AB

rightwards double arrow

F is

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168 Views

Short Answer Type

79. A (3, 2, 0), B (5, 3, 2) and C (-9, 6, -3) are the vertices of a triangle ABC. The bisector AD of ∆ BAC meets BC at D. Find the co-ordinates of D.

1213 Views

80. The co-ordinates of the mid-point C of a line segment AB are (-1, -2, 1). If the co-ordinates of point A are (2, 1, -3), find the co-ordinates of B.

93 Views