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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.

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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.

Let the co-ordinates of the vertices A, B and C respectively be $space space space left parenthesis straight x subscript 1 comma space straight y subscript 1 comma space straight z subscript 1 right parenthesis comma space left parenthesis straight x subscript 2 comma space straight y subscript 2 comma space straight z subscript 2 right parenthesis space and space left parenthesis straight x subscript 3 comma space straight y subscript 3 comma space straight z subscript 3 right parenthesis.$
(2, 1, -1) is mid-point of BC.
∴ $space space open parentheses fraction numerator straight x subscript 2 plus straight x subscript 3 over denominator 2 end fraction comma space fraction numerator straight y subscript 2 plus straight y subscript 3 over denominator 2 end fraction comma space fraction numerator straight z subscript 2 plus straight z subscript 3 over denominator 2 end fraction close parentheses space left right arrow space left parenthesis 2 comma space 1 comma space minus 1 right parenthesis$
$rightwards double arrow$ $straight x subscript 2 plus straight x subscript 3 space equals space 4$ ...(i)
 $straight y subscript 2 plus straight y subscript 3 equals 2$ ...(ii)
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(3, 2, 4) is mid-point of CA.
∴ $space space space space open parentheses fraction numerator straight x subscript 3 plus straight x subscript 1 over denominator 2 end fraction comma space fraction numerator straight y subscript 3 plus straight y subscript 1 over denominator 2 end fraction comma space fraction numerator straight z subscript 3 plus straight z subscript 1 over denominator 2 end fraction close parentheses space left right arrow space left parenthesis 3 comma space 2 comma space 4 right parenthesis$
$rightwards double arrow$ $straight x subscript 3 plus straight x subscript 1 space equals space 6$ ...(iv)
 $space space space straight y subscript 3 plus straight y subscript 1 space equals space 4$ ...(v)
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(1, -2, 3) is mid-point of AB.
∴ $open parentheses fraction numerator straight x subscript 1 plus straight x subscript 2 over denominator 2 end fraction comma space fraction numerator straight y subscript 1 plus straight y subscript 2 over denominator 2 end fraction comma space fraction numerator straight z subscript 1 plus end subscript straight z subscript 2 over denominator 2 end fraction space close parentheses space left right arrow space left parenthesis 1 comma space minus 2 comma space 3 right parenthesis$
$rightwards double arrow$ $straight x subscript 1 plus straight x subscript 2 equals 2$ ...(vii)
 $space space space straight y subscript 1 plus straight y subscript 2 space equals space minus 4$ ...(viii)
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Adding (i), (iv) and (vii), we get $2 straight x subscript 1 plus 2 straight x subscript 2 plus 2 straight x subscript 3 equals 4 plus 6 plus 2 equals 12$
$rightwards double arrow$ $straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 equals 6$ ...(x)
Using (i) in (x), we get $space space space straight x subscript 1 space equals space 2$
Using (iv) in (x), we get $straight x subscript 2 equals 0$
Using (vii) in (x), we get $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
Adding (ii), (v) and (viii), we get $2 straight y subscript 1 plus 2 straight y subscript 2 plus 2 straight y subscript 3 space equals space 2 plus 4 minus 4 equals 2$
$rightwards double arrow$ $straight y subscript 1 plus straight y subscript 2 plus straight y subscript 3 equals 1$ ...(xi)
Using (ii) in (xi), we get $space space straight y subscript 1 equals negative 1$
Using (v) in (xi), we get $space straight y subscript 2 equals negative 3$
Using (viii) in (xi), we get $space straight y subscript 3 equals 5$
Adding (iii), (vi) and (ix), we get $2 straight z subscript 1 plus 2 straight z subscript 2 plus 2 straight z subscript 3 equals negative 2 plus 8 plus 6 equals 12$
$rightwards double arrow$ $straight z subscript 1 plus straight z subscript 2 plus straight z subscript 3 space equals space 6$ ...(xii)
Using (iii) in (xii), we get $space space space space straight z subscript 1 equals 8$
Using (vi) in (xii), we get $straight z subscript 2 equals negative 2$
Using (ix) in (xii), we get $straight z subscript 3 equals 0$
Hence, the three vertices A, B and C are (2, -1, 8), (0, -3, -2), (4, 5, 0) respectively.