{"id":10837,"date":"2012-10-24T00:19:49","date_gmt":"2012-10-23T23:19:49","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=10837"},"modified":"2022-01-20T18:05:26","modified_gmt":"2022-01-20T18:05:26","slug":"um-cubo-com-4-cm-de-aresta","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=10837","title":{"rendered":"Um cubo com $4$ cm de aresta"},"content":{"rendered":"<p><ul id='GTTabs_ul_10837' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_10837' class='GTTabs_curr'><a  id=\"10837_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_10837' ><a  id=\"10837_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_10837'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<div id=\"attachment_10840\" style=\"width: 202px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubopag53.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-10840\" data-attachment-id=\"10840\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10840\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubopag53.jpg\" data-orig-size=\"320,310\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&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\/2012\/10\/cubopag53.jpg\" class=\" wp-image-10840\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubopag53.jpg\" alt=\"\" width=\"192\" height=\"186\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubopag53.jpg 320w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubopag53-300x290.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubopag53-150x145.jpg 150w\" sizes=\"auto, (max-width: 192px) 100vw, 192px\" \/><\/a><p id=\"caption-attachment-10840\" class=\"wp-caption-text\">Cubo com $4$ cm de aresta<\/p><\/div>\n<p>Consideremos o cubo [ABCDEFGH], com $4$ cm de aresta e o plano IJK, sendo J e K pontos m\u00e9dios das arestas [AE] e [DH], respetivamente, e I um ponto de [EF], tal que $\\overline {EI}\u00a0 = 3$ cm.<\/p>\n<ol>\n<li>Qual a posi\u00e7\u00e3o do plano IJK em rela\u00e7\u00e3o \u00e0 reta da aresta [AD]? Porqu\u00ea?<\/li>\n<li>Represente em perspetiva sobre o cubo a sec\u00e7\u00e3o nele produzida pelo plano IJK.<\/li>\n<li>Classifique, justificando, a sec\u00e7\u00e3o obtida e represente-a em verdadeira grandeza.<\/li>\n<li>Determine o valor exato do per\u00edmetro e da \u00e1rea da sec\u00e7\u00e3o.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_10837' onClick='GTTabs_show(1,10837)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_10837'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<div id=\"attachment_10840\" style=\"width: 202px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-10840\" data-attachment-id=\"10840\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10840\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubopag53.jpg\" data-orig-size=\"320,310\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&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\/2012\/10\/cubopag53.jpg\" class=\" wp-image-10840\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubopag53.jpg\" alt=\"\" width=\"192\" height=\"186\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubopag53.jpg 320w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubopag53-300x290.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubopag53-150x145.jpg 150w\" sizes=\"auto, (max-width: 192px) 100vw, 192px\" \/><p id=\"caption-attachment-10840\" class=\"wp-caption-text\">Cubo com $4$ cm de aresta<\/p><\/div>\n<ol>\n<li>As retas JK e AD s\u00e3o paralelas, pois J e K s\u00e3o os pontos m\u00e9dios das arestas [AE] e [DH], respetivamente.<br \/>\nOra, a reta AD \u00e9 paralela \u00e0 reta JK contida no plano IJK. Logo, o plano IJK \u00e9 paralelo \u00e0 reta AD (um plano \u00e9 paralelo a uma reta se nesse plano existir uma reta paralela \u00e0 reta dada).<\/p>\n<\/li>\n<li>A sec\u00e7\u00e3o produzida no cubo pelo plano IJK \u00e9 o quadril\u00e1tero [IJKL].\n<p><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":732,\r\n\"height\":383,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 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 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 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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><\/p>\n<\/li>\n<li>Um plano interseta planos paralelos segundo retas paralelas. Logo, as retas IJ e KL, intersec\u00e7\u00e3o do plano IJK com os planos ABF e DCG, respetivamente, s\u00e3o paralelas.<br \/>\nConsequentemente, os lados [IJ] e [KL] do quadril\u00e1tero s\u00e3o paralelos.<\/p>\n<p>Esses lados s\u00e3o geometricamente iguais, pois s\u00e3o hipotenusas dos tri\u00e2ngulos ret\u00e2ngulos [IEJ] e [LHK] geometricamente iguais.<\/p>\n<p>Como a reta JK \u00e9 perpendicular ao plano ABF (pois \u00e9 paralela \u00e0 reta AD), ent\u00e3o \u00e9 perpendicular a todas as retas desse plano e, em particular, \u00e0 reta IJ. Consequentemente, \u00e9 reto o \u00e2ngulo IJK.<\/p>\n<p>Assim, conclui-se que o quadril\u00e1tero [IJKL] \u00e9 um ret\u00e2ngulo, sendo $\\overline {JK}\u00a0 = \\overline {IL}\u00a0 = 4$ cm e $\\overline {IJ}\u00a0 = \\overline {KL}\u00a0 = \\sqrt {{2^2} + {3^2}}\u00a0 = \\sqrt {13} $ cm.<\/p>\n<\/li>\n<li>A sec\u00e7\u00e3o produzida no cubo tem de per\u00edmetro $P = 2 \\times \\left( {4 + \\sqrt {13} } \\right) = 8 + 2\\sqrt {13} $ cm e de \u00e1rea $A = 4 \\times \\sqrt {13}\u00a0 = 4\\sqrt {13} $ cm<sup>2<\/sup>.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_10837' onClick='GTTabs_show(0,10837)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Consideremos o cubo [ABCDEFGH], com $4$ cm de aresta e o plano IJK, sendo J e K pontos m\u00e9dios das arestas [AE] e [DH], respetivamente, e I um ponto de [EF],&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20755,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,334],"tags":[429,67,333],"series":[],"class_list":["post-10837","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-10-o-ano","category-aplicando","category-resolucao-de-problemas-de-geometria-no-plano-e-no-espaco","tag-10-o-ano","tag-geometria","tag-seccoes"],"views":3812,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/10V1Pag063-1_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10837","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=10837"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10837\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20755"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10837"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10837"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10837"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=10837"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}