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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.

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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.

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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.

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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).

Let R and S be the two points of trisection:
Ratio at R is 1 : 2
$rightwards double arrow$ $straight x equals fraction numerator 1 left parenthesis 1 right parenthesis plus 2 left parenthesis 3 right parenthesis over denominator 1 plus 2 end fraction equals 7 over 3$

   $straight y equals fraction numerator 1 left parenthesis 2 right parenthesis plus 2 left parenthesis 2 right parenthesis over denominator 1 plus 2 end fraction equals 6 over 3 equals 2$

   $space space straight z equals fraction numerator 1 left parenthesis 5 right parenthesis plus 2 left parenthesis negative 1 right parenthesis over denominator 1 plus 2 end fraction equals 3 over 3 equals 1$

∴ Point R is $space space open parentheses 7 over 3 comma space 2 comma space 1 close parentheses$
Also, ratio at S is 2 : 1
$rightwards double arrow$ $straight x equals fraction numerator 2 left parenthesis 1 right parenthesis plus 1 left parenthesis 3 right parenthesis over denominator 2 plus 1 end fraction equals 5 over 3$

$straight y equals fraction numerator 2 left parenthesis 2 right parenthesis plus 1 left parenthesis 2 right parenthesis over denominator 2 plus 1 end fraction equals 6 over 3 equals 2$

   $straight z equals fraction numerator 2 left parenthesis 5 right parenthesis plus 1 left parenthesis negative 1 right parenthesis over denominator 2 plus 1 end fraction equals 9 over 3 equals 3$
∴ Point S is $open parentheses 5 over 3 comma space 2 comma space 3 close parentheses$

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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).

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76.

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

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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.

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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).

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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.

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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.

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