{"id":6367,"date":"2010-12-12T18:44:33","date_gmt":"2010-12-12T18:44:33","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6367"},"modified":"2022-01-12T15:21:00","modified_gmt":"2022-01-12T15:21:00","slug":"dados-tres-pontos","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6367","title":{"rendered":"Dados tr\u00eas pontos"},"content":{"rendered":"<p><ul id='GTTabs_ul_6367' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6367' class='GTTabs_curr'><a  id=\"6367_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6367' ><a  id=\"6367_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_6367'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Sejam $A\\,(2,0,0,)$, $B\\,(-4,0,0)$ e $C\\,(0,6,0)$ tr\u00eas pontos dados pelas suas coordenadas num referencial ortonormado.<\/p>\n<ol>\n<li>Determine as equa\u00e7\u00f5es dos planos mediadores dos segmentos de reta [AB], [BC] e [CA].<\/li>\n<li>Mostre que estes planos t\u00eam uma reta comum e indique uma equa\u00e7\u00e3o desta reta.<\/li>\n<li>Determine as coordenadas do ponto D de modo que [ABCD] seja um paralelogramo.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6367' onClick='GTTabs_show(1,6367)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6367'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Sejam $A\\,(2,0,0,)$, $B\\,(-4,0,0)$ e $C\\,(0,6,0)$ tr\u00eas pontos dados pelas suas coordenadas num referencial ortonormado.<\/p>\n<\/blockquote>\n<ol>\n<li>Os pontos m\u00e9dios dos segmentos considerados s\u00e3o:<br \/>\n\\[\\begin{array}{*{20}{l}}{{M_{[AB]}} = ( &#8211; 1,0,0)}&amp;,&amp;{{M_{[BC]}} = ( &#8211; 2,3,0)}&amp;e&amp;{{M_{[CA]}} = (1,3,0)}\\end{array}\\]<br \/>\nDesignado por $P\\,(x,y,z)$ um ponto gen\u00e9rico do plano mediador de [AB], ter-se-\u00e1 $\\overrightarrow{{{M}_{[AB]}}P}.\\overrightarrow{AB}=0$.<br \/>\n(Verifique geometricamente, atrav\u00e9s de um desenho)<\/p>\n<p>Assim, vem:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{{{M}_{[AB]}}P}.\\overrightarrow{AB}=0 &amp; \\Leftrightarrow\u00a0 &amp; (x+1,y,z).(-6,0,0)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; -6x-6=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; x=-1\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nPortanto, $x=-1$ \u00e9 uma equa\u00e7\u00e3o do plano mediador de [AB].<\/p>\n<p>De modo semelhante, temos:<\/p>\n<p>\\[\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{{{M}_{[BC]}}P}.\\overrightarrow{BC}=0 &amp; \\Leftrightarrow\u00a0 &amp; (x+2,y-3,z).(4,6,0)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 4x+8+6y-18=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 2x+3y-5=0\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nPortanto, $2x+3y-5=0$ \u00e9 uma equa\u00e7\u00e3o do plano mediador de [BC].<\/p>\n<p>\\[\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{{{M}_{[CA]}}P}.\\overrightarrow{CA}=0 &amp; \\Leftrightarrow\u00a0 &amp; (x-1,y-3,z).(2,-6,0)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 2x-2-6y+18=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; x-3y+8=0\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nPortanto, $x-3y+8=0$ \u00e9 uma equa\u00e7\u00e3o do plano mediador de [CA].<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Designemos, pela ordem acima indicada, os planos por $\\alpha $, $\\beta $ e $\\gamma $.<\/p>\n<p>Os vetores $\\overrightarrow{{{n}_{\\alpha }}}(1,0,0)$, $\\overrightarrow{{{n}_{\\beta }}}(2,3,0)$ e $\\overrightarrow{{{n}_{\\gamma }}}(1,-3,0)$ s\u00e3o, respetivamente, vetores normais a estes planos.<\/p>\n<p>Como os vetores\u00a0\u00a0$\\overrightarrow{{{n}_{\\alpha }}}(1,0,0)$ e $\\overrightarrow{{{n}_{\\beta }}}(2,3,0)$ s\u00e3o obl\u00edquos, ent\u00e3o os respectivos planos s\u00e3o tamb\u00e9m obl\u00edquos. Determinemos uma equa\u00e7\u00e3o da reta de intersec\u00e7\u00e3o destes planos:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=-1\u00a0 \\\\<br \/>\n2x+3y-5=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=-1\u00a0 \\\\<br \/>\ny=\\frac{7}{3}\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nPortanto, $x=-1\\wedge y=\\frac{7}{3}$ \u00e9 uma equa\u00e7\u00e3o da reta de intersec\u00e7\u00e3o dos planos $\\alpha $ e $\\beta $.<\/p>\n<p>Dois pontos desta reta s\u00e3o, por exemplo, $S\\,(-1,\\frac{7}{3},0)$ e $T\\,(-1,\\frac{7}{3},1)$.<\/p>\n<p>Confirmemos, seguidamente, que as coordenadas destes pontos verificam a equa\u00e7\u00e3o do plano\u00a0\u00a0${\\gamma }$.<\/p>\n<p>Ora, $-1-3\\times \\frac{7}{3}+8=0\\Leftrightarrow -8+8=0$ \u00e9 uma proposi\u00e7\u00e3o verdadeira.<br \/>\nLogo, os pontos S e T pertencem ao plano\u00a0\u00a0${\\gamma }$.<\/p>\n<p>Sendo assim, a reta de intersec\u00e7\u00e3o dos planos\u00a0\u00a0$\\alpha $ e $\\beta $ \u00e9 uma reta contida no plano ${\\gamma }$.<\/p>\n<p>Consequentemente, os tr\u00eas planos t\u00eam uma reta em comum, de equa\u00e7\u00e3o $x=-1\\wedge y=\\frac{7}{3}$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>O ponto pedido pode ser obtido pela rela\u00e7\u00e3o $D=C+\\overrightarrow{BA}$. (Porqu\u00ea?)<br \/>\nLogo, $D=(0,6,0)+(6,0,0)=(6,6,0)$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6367' onClick='GTTabs_show(0,6367)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sejam $A\\,(2,0,0,)$, $B\\,(-4,0,0)$ e $C\\,(0,6,0)$ tr\u00eas pontos dados pelas suas coordenadas num referencial ortonormado. Determine as equa\u00e7\u00f5es dos planos mediadores dos segmentos de reta [AB], [BC] e [CA]. Mostre que estes&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14083,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67,119],"series":[],"class_list":["post-6367","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-interseccao-de-planos"],"views":2362,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Mat28.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6367","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=6367"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6367\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14083"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6367"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6367"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6367"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}