{"id":6377,"date":"2010-12-13T21:41:10","date_gmt":"2010-12-13T21:41:10","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6377"},"modified":"2022-01-21T22:39:08","modified_gmt":"2022-01-21T22:39:08","slug":"outro-cubo-abcdefgh","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6377","title":{"rendered":"Outro cubo [ABCDEFGH]"},"content":{"rendered":"<p><ul id='GTTabs_ul_6377' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6377' class='GTTabs_curr'><a  id=\"6377_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6377' ><a  id=\"6377_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_6377'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6378\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6378\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50.jpg\" data-orig-size=\"341,404\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Cubo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50.jpg\" class=\"alignright wp-image-6378\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50-253x300.jpg\" alt=\"\" width=\"270\" height=\"320\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50-253x300.jpg 253w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50-126x150.jpg 126w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50.jpg 341w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/a>Seja $(O,\\vec{i},\\vec{j},\\vec{k})$ um referencial o.n. do espa\u00e7o.<\/p>\n<p>A figura representa um cubo [ABCDEFGH] de centro O e aresta 2 cm, sendo as arestas [AD] e [DC] paralelas a Ox e Oy, respetivamente.<\/p>\n<p>Os pontos I, J, K, L, M, N, P e Q s\u00e3o pontos m\u00e9dios das arestas.<\/p>\n<ol>\n<li>Determine as coordenadas dos pontos anteriormente referidos.<\/li>\n<li>Indique as coordenadas dos pontos sim\u00e9tricos de K, G e E em rela\u00e7\u00e3o a J.<\/li>\n<li>Defina, por uma condi\u00e7\u00e3o, o plano da face [ADHE].<\/li>\n<li>Escreva equa\u00e7\u00f5es das retas NK, IU e IK.<\/li>\n<li>Escreva equa\u00e7\u00f5es da recta paralela a QP e que passa por I.<\/li>\n<li>Prove que a diagonal [FD] do cubo \u00e9 perpendicular ao plano que cont\u00e9m o hex\u00e1gono.<\/li>\n<li>Calcule a \u00e1rea do hex\u00e1gono [IMNKPQ].<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6377' onClick='GTTabs_show(1,6377)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6377'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6378\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6378\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50.jpg\" data-orig-size=\"341,404\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Cubo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50.jpg\" class=\"alignright wp-image-6378\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50-253x300.jpg\" alt=\"\" width=\"270\" height=\"320\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50-253x300.jpg 253w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50-126x150.jpg 126w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-50.jpg 341w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/a>As coordenadas desses pontos s\u00e3o: $I\\,(1,-1,0)$, $J\\,(1,1,0)$, $K\\,(-1,1,0)$, $L\\,(-1,-1,0)$, $M\\,(1,0,-1)$, $N\\,(0,1,-1)$, $P\\,(-1,0,1)$ e $Q\\,(0,-1,1)$.<br \/>\n\u00ad<\/li>\n<li>As coordenadas desses pontos s\u00e3o, respetivamente, $K&#8217;\\,(3,1,0)$, $G&#8217;\\,(3,1,-1)$ e $E&#8217;\\,(1,3,-1)$.<br \/>\n\u00ad<\/li>\n<li>\u00a0O plano da face [ADHE] pode ser definido pela equa\u00e7\u00e3o $y=-1$.<br \/>\n\u00ad<\/li>\n<li>Por visualiza\u00e7\u00e3o, podemos definir essas retas por:<br \/>\nNK: $y=1\\wedge z=-x-1$<br \/>\nIJ: $x=1\\wedge z=0$<br \/>\nIK: $y=-x\\wedge z=0$<\/p>\n<p><span style=\"text-decoration: underline;\">Confirmando NK<\/span>:<br \/>\nComo $N\\,(0,1,-1)$ e $K\\,(-1,1,0)$, ent\u00e3o $\\overrightarrow{NK}=(-1,0,1)$.<br \/>\nLogo, vem: $\\frac{x-0}{-1}=\\frac{z+1}{1}\\wedge y=1\\Leftrightarrow y=1\\wedge z=-x-1$.<\/p>\n<p><span style=\"text-decoration: underline;\">Confirmando IK<\/span>:<br \/>\nComo $I\\,(1,-1,0)$ e $K\\,(-1,1,0)$, ent\u00e3o $\\overrightarrow{IK}=(-2,2,0)$.<br \/>\nLogo, vem: $\\frac{x-1}{-2}=\\frac{y+1}{2}\\wedge z=0\\Leftrightarrow -x+1=y+1\\wedge z=0\\Leftrightarrow y=-x\\wedge z=0$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>A reta considerada \u00e9 a reta IK, que pode ser definida por $y=-x\\wedge z=0$, como vimos na al\u00ednea anterior.<br \/>\n\u00ad<\/li>\n<li>Para isso basta provar que $\\overrightarrow{DF}\\bot \\overrightarrow{MI}\\wedge \\overrightarrow{DF}\\bot \\overrightarrow{MN}$.\n<p>Ora, $\\overrightarrow{DF}=(2,2,2)$, $\\overrightarrow{MI}=(0,-1,1)$ e $\\overrightarrow{MN}=(-1,1,0)$.<\/p>\n<p>Donde, $\\overrightarrow{DF}.\\overrightarrow{MI}=2\\times 0+2\\times (-1)+2\\times 1=0$ e $\\overrightarrow{DF}.\\overrightarrow{MN}=2\\times (-1)+2\\times 1+2\\times 0=0$.<\/p>\n<p>Portanto, a diagonal [FD] \u00e9 perpendicular ao plano que cont\u00e9m o hex\u00e1gono, pois \u00e9 perpendicular a duas retas (MI e MN) concorrentes desse plano.<\/p>\n<\/li>\n<li>O hex\u00e1gono \u00e9 regular, com lado igual a metade da diagonal facial do cubo, isto \u00e9 $l=\\overline{PQ}=\\frac{2\\sqrt{2}}{2}=\\sqrt{2}$.\n<p>Consideremos o hex\u00e1gono decomposto em seis tri\u00e2ngulos equil\u00e1teros geometricamente iguais.<\/p>\n<p>A altura desse tri\u00e2ngulo equil\u00e1tero \u00e9 \\[h=\\sqrt{{{\\left( \\sqrt{2} \\right)}^{2}}-{{\\left( \\frac{\\sqrt{2}}{2} \\right)}^{2}}}=\\sqrt{2-\\frac{1}{2}}=\\sqrt{\\frac{3}{2}}\\]<br \/>\nLogo, a \u00e1rea do hex\u00e1gono \u00e9: \\[A=6\\times \\frac{\\sqrt{2}\\times \\sqrt{\\frac{3}{2}}}{2}=3\\sqrt{3}\\,c{{m}^{2}}\\]<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6377' onClick='GTTabs_show(0,6377)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Seja $(O,\\vec{i},\\vec{j},\\vec{k})$ um referencial o.n. do espa\u00e7o. A figura representa um cubo [ABCDEFGH] de centro O e aresta 2 cm, sendo as arestas [AD] e [DC] paralelas a Ox e Oy,&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20837,"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],"series":[],"class_list":["post-6377","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"],"views":2193,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag186-50_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6377","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=6377"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6377\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20837"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6377"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6377"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6377"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6377"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}