{"id":5599,"date":"2010-11-17T00:49:43","date_gmt":"2010-11-17T00:49:43","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5599"},"modified":"2022-01-12T12:24:04","modified_gmt":"2022-01-12T12:24:04","slug":"determine-uma-equacao-cartesiana-do-plano-mediador-do-segmento-ab","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5599","title":{"rendered":"Determine uma equa\u00e7\u00e3o cartesiana do plano mediador do segmento [AB]"},"content":{"rendered":"<p><ul id='GTTabs_ul_5599' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5599' class='GTTabs_curr'><a  id=\"5599_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5599' ><a  id=\"5599_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_5599'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Determine uma equa\u00e7\u00e3o cartesiana do plano mediador do segmento de reta [AB], sendo:<\/p>\n<ol>\n<li>$A(4,-1,2)$ e $B(2,7,0)$.<\/li>\n<li>$A(-4,1,7)$ e $B(3,2,-5)$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5599' onClick='GTTabs_show(1,5599)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5599'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><span class=\"alignright\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"text-align: right\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 1,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 0,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/span>O plano mediador do segmento de reta [AB] \u00e9 o lugar geom\u00e9trico dos pontos $P(x,y,z)$ do espa\u00e7o, tais que $\\overrightarrow{MP}.\\overrightarrow{AB}=0$, sendo M o ponto m\u00e9dio de [AB].\n<p>Ora,\u00a0$M(\\frac{4+2}{2},\\frac{-1+7}{2},\\frac{2+0}{2})=(3,3,1)$.<\/p>\n<p>Assim, vem:<\/p>\n<p>$\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{MP}.\\overrightarrow{AB}=0 &amp; \\Leftrightarrow\u00a0 &amp; (x-3,y-3,z-1).(-2,8,-2)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; -2x+6+8y-24-2z+2=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; -2x+8y-2z-16=0\u00a0 \\\\<br \/>\n\\end{array}$<\/p>\n<p>Logo, $-2x+8y-2z-16=0$ \u00e9 uma equa\u00e7\u00e3o cartesiana do plano mediador de [AB].<br \/>\n\u00ad<\/p>\n<\/li>\n<li>O plano mediador so segmento [AB] \u00e9 o lugar geom\u00e9trico dos pontos $P(x,y,z)$ do espa\u00e7o, tais que $\\overrightarrow{MP}.\\overrightarrow{AB}=0$, sendo M o ponto m\u00e9dio de [AB].\n<p>Ora,\u00a0$M(\\frac{-4+3}{2},\\frac{1+2}{2},\\frac{7-5}{2})=(-\\frac{1}{2},\\frac{3}{2},1)$.<\/p>\n<p>Assim, vem:<\/p>\n<p>$\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{MP}.\\overrightarrow{AB}=0 &amp; \\Leftrightarrow\u00a0 &amp; (x+\\frac{1}{2},y-\\frac{3}{2},z-1).(7,1,-12)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 7x+\\frac{7}{2}+y-\\frac{3}{2}-12z+12=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 7x+y-12z+14=0\u00a0 \\\\<br \/>\n\\end{array}$<\/p>\n<p>Logo,\u00a0$7x+y-12z+14=0$ \u00e9 uma equa\u00e7\u00e3o cartesiana do plano mediador de [AB].<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5599' onClick='GTTabs_show(0,5599)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Determine uma equa\u00e7\u00e3o cartesiana do plano mediador do segmento de reta [AB], sendo: $A(4,-1,2)$ e $B(2,7,0)$. $A(-4,1,7)$ e $B(3,2,-5)$. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19403,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67,111],"series":[],"class_list":["post-5599","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-geometria-analitica","tag-11-o-ano","tag-geometria","tag-vectores"],"views":39966,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/Plano_mediador_de_um_segmento_de_reta.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5599","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5599"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5599\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19403"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5599"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5599"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5599"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5599"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}