Show that the points:A (-2, 6, -2), B (0, 4, -1), C (-2, 3, 1) a
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    Introduction To Three Dimensional Geometry

    Multiple Choice QuestionsShort Answer Type

    31. Verify the following : (0, 7, -10), (1, 6, -6) and ( 4, 9, -6) are the vertices of an isosceles triangle.
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    32. Show that A (a, b, c), B (b, c, a) and C (c, a, b) are vertices of an equilateral triangle.
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    33. Show that A (6, 10, 10), B (1, 0, -5) and C (6, -10, 0) are the vertices of a right angled triangle.
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    34.

    Verify the following:
    (0, 7, 10), (-1, 6, 6) and (-4, 9, 6) are the vertices of a right angled triangle. 

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    Multiple Choice QuestionsLong Answer Type

    35. Verify the following:
    (-1, 2, 1), (1, -2, 5), (4, -7, 8) and (2, -3, 4) are the vertices of a parallelogram.
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    36. Verify the following:
    Show that A (6, 10, 10), B (1, 0, -5), C (6, -10, 0) and D (11, 0, 15) are the vertices of a rectangle.
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    37. Show that points A (1, 2, 3), B (-1, -2, -1), C (2, 3, 2) and D (4, 7, 6) are the vertices of a parallelogram ABCD, but it is not a rectangle.
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    Multiple Choice QuestionsShort Answer Type

    38.

    Show that the points:
    P (-1,-6, 10), Q (1, -3, 4), R (-5, -1, 1) and S (-7, -4, 7) are the vertices of a rhombus.

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    Multiple Choice QuestionsLong Answer Type

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    39. Show that the points:
    A (-2, 6, -2), B (0, 4, -1), C (-2, 3, 1) and D (-4, 5, 0) are the vertices of a square.

    straight A space left parenthesis negative 2 comma space 6 comma space minus 2 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 minus 1 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 negative 2 comma space 3 comma space 1 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 space straight D space left parenthesis negative 4 comma space 5 comma space 0 right parenthesis space left right arrow space left parenthesis straight x subscript 4 comma space straight y subscript 4 comma space straight z subscript 4 right parenthesis
    Using distance formula, we have   
                         space space space space space space space AB equals space square root of left parenthesis straight x subscript 1 minus straight x subscript 2 right parenthesis squared plus left parenthesis straight y subscript 1 minus straight y subscript 2 right parenthesis squared plus left parenthesis straight z subscript 1 minus straight z subscript 2 right parenthesis squared end root
                   <pre>uncaught exception: <b>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</b><br /><br />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<br />#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>                  ...(i)

        <pre>uncaught exception: <b>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</b><br /><br />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<br />#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>
             equals space square root of left parenthesis 0 plus 2 right parenthesis squared plus left parenthesis 4 minus 3 right parenthesis squared plus left parenthesis negative 1 minus 1 right parenthesis squared end root space equals space square root of 4 plus 1 plus 4 end root space equals space square root of 9 space equals space 3                            ...(ii)

    <pre>uncaught exception: <b>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</b><br /><br />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<br />#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>
             equals space square root of left parenthesis negative 2 plus 4 right parenthesis squared plus left parenthesis 3 minus 5 right parenthesis squared plus left parenthesis 1 minus 0 right parenthesis squared end root space equals space square root of 4 plus 4 plus 1 end root space equals space square root of 9 space equals space 3                           ....(iii)

       DA space equals space square root of left parenthesis straight x subscript 4 minus straight x subscript 1 right parenthesis squared plus left parenthesis straight y subscript 4 minus straight y subscript 1 right parenthesis squared plus left parenthesis straight z subscript 4 minus straight z subscript 1 right parenthesis squared end root
          space space space space equals space square root of left parenthesis negative 4 plus 2 right parenthesis squared plus left parenthesis 5 minus 6 right parenthesis squared plus left parenthesis 0 plus 2 right parenthesis squared end root space equals space square root of 4 plus 1 plus 4 end root space equals space square root of 9 space equals space 3                            ...(iv)

    space space space AC space equals space square root of left parenthesis straight x subscript 1 minus straight x subscript 3 right parenthesis squared plus left parenthesis straight y subscript 1 minus straight y subscript 3 right parenthesis squared plus left parenthesis straight z subscript 1 minus straight z subscript 3 right parenthesis squared end root
       space space space space space space equals space square root of left parenthesis negative 2 plus 2 right parenthesis squared plus left parenthesis 6 minus 3 right parenthesis squared plus left parenthesis negative 2 minus 1 right parenthesis squared end root space equals space square root of 0 plus 9 plus 9 end root space equals space square root of 18                               ....(v)

      BD space equals space square root of left parenthesis straight x subscript 2 minus straight x subscript 4 right parenthesis squared plus left parenthesis straight y subscript 2 minus straight y subscript 4 right parenthesis squared plus left parenthesis straight z subscript 2 minus straight z subscript 4 right parenthesis squared end root
          space space space equals space square root of left parenthesis 0 plus 4 right parenthesis squared plus left parenthesis 4 minus 5 right parenthesis squared plus left parenthesis negative 1 minus 0 right parenthesis squared end root space equals space square root of 16 plus 1 plus 1 end root space equals space square root of 18                                ...(vi)
    From (i), (ii), (iii), (iv), (v) and (vi), we have
                     AB = BC = CD = DA and AC = BD
    Hence, ABCD is a square.
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    Multiple Choice QuestionsShort Answer Type

    40. Find the co-ordinates of a point equidistant from points A (a, 0, 0), B (0, b, 0), C (0, 0, c) and O (0, 0, 0).
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