{"id":5563,"date":"2010-11-16T19:03:14","date_gmt":"2010-11-16T19:03:14","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5563"},"modified":"2022-01-12T12:18:21","modified_gmt":"2022-01-12T12:18:21","slug":"identifique-o-conjunto-de-pontos-do-plano-definidos-pela-condicao","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5563","title":{"rendered":"Identifique o conjunto de pontos do plano definidos pela condi\u00e7\u00e3o"},"content":{"rendered":"<p><ul id='GTTabs_ul_5563' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5563' class='GTTabs_curr'><a  id=\"5563_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5563' ><a  id=\"5563_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_5563'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Sendo $A(0,9)$ e $B(-8,2)$, identifique o conjunto de pontos $P(x,y)$ do plano que verificam a condi\u00e7\u00e3o:<\/p>\n<ol>\n<li>$\\overrightarrow{AP}.\\overrightarrow{BP}=0$;<\/li>\n<li>$\\overrightarrow{MP}.\\overrightarrow{AM}=0$, sendo M o ponto m\u00e9dio de [AB].<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5563' onClick='GTTabs_show(1,5563)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5563'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Tente identificar o lugar geom\u00e9trico definido pela condi\u00e7\u00e3o $\\overrightarrow{AP}.\\overrightarrow{BP}=0$.<br \/>\nCaso n\u00e3o consiga, execute a anima\u00e7\u00e3o sem ativar &#8220;Mostrar lugar geom\u00e9trico&#8221;.<br \/>\nDe seguida, verifique a sua suposi\u00e7\u00e3o.<\/p>\n<p><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"float: right;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":267,\r\n\"height\":296,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49  50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ 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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': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'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><\/p>\n<p>O conjunto dos pontos $P(x,y)$ do plano que verificam a condi\u00e7\u00e3o $\\overrightarrow{AP}.\\overrightarrow{BP}=0$ \u00e9 a circunfer\u00eancia de di\u00e2metro [AB].<\/p>\n<p>Com efeito,<\/p>\n<p>$\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{AP}.\\overrightarrow{BP}=0 &amp; \\Leftrightarrow\u00a0 &amp; (x,y-9).(x+8,y-2)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{x}^{2}}+8x+{{y}^{2}}-11y+18=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{(x+4)}^{2}}-16+{{(y-\\frac{11}{2})}^{2}}-\\frac{121}{4}+18=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{(x+4)}^{2}}+{{(y-\\frac{11}{2})}^{2}}=\\frac{113}{4}\u00a0 \\\\<br \/>\n\\end{array}$<\/p>\n<p>Ora:<\/p>\n<p>o centro da circunfer\u00eancia \u00e9 o ponto m\u00e9dio de [AB], ponto $C(\\frac{0-8}{2},\\frac{9+2}{2})=(-4,\\frac{11}{2})$;<\/p>\n<p>o raio a circunfer\u00eancia \u00e9 $r=\\frac{\\overline{AB}}{2}=\\frac{1}{2}\\sqrt{{{8}^{2}}+{{7}^{2}}}=\\frac{\\sqrt{113}}{2}$.<\/p>\n<\/li>\n<li>\n<p>Tente identificar o lugar geom\u00e9trico definido pela condi\u00e7\u00e3o $\\overrightarrow{MP}.\\overrightarrow{AM}=0$, sendo M o ponto m\u00e9dio de [AB].<br \/>\nCaso n\u00e3o consiga, execute a anima\u00e7\u00e3o sem ativar &#8220;Mostrar lugar geom\u00e9trico&#8221;.<br \/>\nDe seguida, verifique a sua suposi\u00e7\u00e3o.<\/p>\n<p><div id=\"ggbApplet2\" style=\"float: right;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": 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\\Leftrightarrow\u00a0 &amp; 7y=-8x+\\frac{13}{2}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; y=-\\frac{8}{7}x+\\frac{13}{14}\u00a0 \\\\<br \/>\n\\end{array}$<\/p>\n<p>Ora:<\/p>\n<p>o ponto M pertence a esta reta: $\\frac{11}{2}=-\\frac{8}{7}\\times (-4)+\\frac{13}{14}\\Leftrightarrow \\frac{77}{14}=\\frac{64}{14}+\\frac{13}{14}$ (P.V.);<\/p>\n<p>esta reta \u00e9 perpendicular \u00e0 recta AB, pois o declive de uma delas \u00e9 sim\u00e9trico do inverso do da outra.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5563' onClick='GTTabs_show(0,5563)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sendo $A(0,9)$ e $B(-8,2)$, identifique o conjunto de pontos $P(x,y)$ do plano que verificam a condi\u00e7\u00e3o: $\\overrightarrow{AP}.\\overrightarrow{BP}=0$; $\\overrightarrow{MP}.\\overrightarrow{AM}=0$, sendo M o ponto m\u00e9dio de [AB]. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19188,"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-5563","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":6330,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat74.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5563","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=5563"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5563\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19188"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5563"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5563"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5563"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5563"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}