Sponsor Area

Sequences And Series

Question
CBSEENMA11014978

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.

Solution
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
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#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)
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#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.