{"id":10883,"date":"2012-10-24T23:22:13","date_gmt":"2012-10-24T22:22:13","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=10883"},"modified":"2022-01-20T18:13:43","modified_gmt":"2022-01-20T18:13:43","slug":"considere-o-cubo-com-4-cm-de-aresta-representado-na-figura","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=10883","title":{"rendered":"Considere o cubo com $4$ cm de aresta representado na figura"},"content":{"rendered":"<p><ul id='GTTabs_ul_10883' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_10883' class='GTTabs_curr'><a  id=\"10883_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_10883' ><a  id=\"10883_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_10883'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<div id=\"attachment_10885\" style=\"width: 171px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo-pag63-3.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-10885\" data-attachment-id=\"10885\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10885\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo-pag63-3.jpg\" data-orig-size=\"269,255\" 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\/cubo-pag63-3.jpg\" class=\" wp-image-10885\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo-pag63-3.jpg\" alt=\"\" width=\"161\" height=\"153\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo-pag63-3.jpg 269w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo-pag63-3-150x142.jpg 150w\" sizes=\"auto, (max-width: 161px) 100vw, 161px\" \/><\/a><p id=\"caption-attachment-10885\" class=\"wp-caption-text\">Cubo com $4$ cm de aresta<\/p><\/div>\n<p>Consideremos o cubo com $4$ cm de aresta representado na figura.<\/p>\n<p>Sabendo que I e J s\u00e3o pontos m\u00e9dios das arestas a que pertencem:<\/p>\n<ol>\n<li>reproduza o cubo e construa a sec\u00e7\u00e3o nele produzida pelo plano IDJ;<\/li>\n<li>prove\u00a0que a sec\u00e7\u00e3o obtida na al\u00ednea anterior \u00e9 um losango e represente-a em verdadeira grandeza;<\/li>\n<li>determine os valores exatos do per\u00edmetro e da \u00e1rea da sec\u00e7\u00e3o, apresentando o resultado o mais simplificado poss\u00edvel.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_10883' onClick='GTTabs_show(1,10883)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_10883'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-3.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"10889\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10889\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-3.png\" data-orig-size=\"875,445\" 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\/Cubo-pag63-3.png\" class=\"aligncenter  wp-image-10889\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-3.png\" alt=\"\" width=\"700\" height=\"356\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-3.png 875w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-3-300x152.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-3-150x76.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-3-400x203.png 400w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/a><\/p>\n<\/p>\n<ol>\n<li>A sec\u00e7\u00e3o produzida no cubo pelo plano IDJ \u00e9 o quadril\u00e1tero [IDJF], cuja constru\u00e7\u00e3o est\u00e1 apresentada na figura acima.<br \/>\n\u00ad<\/li>\n<li>O quadril\u00e1tero [IDJF] possui os lados opostos paralelos, pois o plano IDJ interseta planos paralelos (que cont\u00eam faces opostas do cubo) segundo retas paralelas.\n<p>Por outro lado, o quadril\u00e1tero possui os quatro lados geometricamente iguais, pois s\u00e3o hipotenusas de tri\u00e2ngulos ret\u00e2ngulos geometricamente iguais ao tri\u00e2ngulo [IEF], sendo (em cm)<br \/>\n$$\\overline {IF}\u00a0 = \\overline {FJ}\u00a0 = \\overline {JD}\u00a0 = \\overline {DI}\u00a0 = \\sqrt {{2^2} + {4^2}}\u00a0 = \\sqrt {20}\u00a0 = 2\\sqrt 5 $$<\/p>\n<p>Portanto, o quadril\u00e1tero [IDJF] possui os quatro lados geometricamente iguais e paralelos dois a dois. Por isso, o quadril\u00e1tero \u00e9 um losango, propriamente losango ou quadrado.<\/p>\n<p>Contudo, como as suas diagonais s\u00e3o diferentes, pois uma \u00e9 uma diagonal espacial do cubo ($\\overline {FD}\u00a0 = 4\\sqrt 3 $ cm) e a outra \u00e9 geometricamente igual a uma diagonal facial do cubo ($\\overline {IJ}\u00a0 = 4\\sqrt 2 $ cm), conclui-se que \u00e9 um losango n\u00e3o quadrado.<\/p>\n<p>A sec\u00e7\u00e3o obtida est\u00e1 representada em verdadeira grandeza na figura acima.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>A sec\u00e7\u00e3o produzida no cubo tem de per\u00edmetro $P = 4 \\times \\overline {IF}\u00a0 = 4 \\times 2\\sqrt 5\u00a0 = 8\\sqrt 5 $ cm e de \u00e1rea $A = \\frac{{\\overline {IJ}\u00a0 \\times \\overline {FD} }}{2} = \\frac{{4\\sqrt 2\u00a0 \\times 4\\sqrt {3\\;} }}{2} = 8\\sqrt 6 $ cm<sup>2<\/sup>.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_10883' onClick='GTTabs_show(0,10883)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Consideremos o cubo com $4$ cm de aresta representado na figura. Sabendo que I e J s\u00e3o pontos m\u00e9dios das arestas a que pertencem: reproduza o cubo e construa a sec\u00e7\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20757,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,334],"tags":[429,324,67,335,333],"series":[],"class_list":["post-10883","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-areas","tag-geometria","tag-perimetro","tag-seccoes"],"views":4966,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/10V1Pag063-3_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10883","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=10883"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10883\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20757"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10883"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10883"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10883"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=10883"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}