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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

Here, A ((3, -5, 7)) and B (-1, 7, -6) be the given vertices of a $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
Let C be (x, y, z)
$rightwards double arrow$ Centroid of triangle ABC is $straight G open parentheses fraction numerator 3 plus left parenthesis negative 1 right parenthesis plus straight x over denominator 3 end fraction comma space fraction numerator left parenthesis negative 5 right parenthesis plus 7 plus straight y over denominator 3 end fraction comma space fraction numerator 7 plus left parenthesis negative 6 right parenthesis plus straight z over denominator 3 end fraction close parentheses space left right arrow space left parenthesis 1 comma space 1 comma space 1 right parenthesis$ (Given)
$rightwards double arrow$ $fraction numerator straight x plus 2 over denominator 3 end fraction equals 1 space or space straight x space equals space 1$
$fraction numerator straight y plus 2 over denominator 3 end fraction equals 1 space or space straight y space equals space 1$
$fraction numerator straight z plus 1 over denominator 3 end fraction equals 1 space or space straight z space equals space 2$
Hence, vertex C is (1, 1, 2).

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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.

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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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