{"id":10861,"date":"2012-10-24T02:23:58","date_gmt":"2012-10-24T01:23:58","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=10861"},"modified":"2022-01-20T18:09:00","modified_gmt":"2022-01-20T18:09:00","slug":"o-mesmo-cubo-com-4-cm-de-aresta","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=10861","title":{"rendered":"O mesmo cubo com $4$ cm de aresta"},"content":{"rendered":"<p><ul id='GTTabs_ul_10861' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_10861' class='GTTabs_curr'><a  id=\"10861_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_10861' ><a  id=\"10861_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_10861'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere, ainda, o cubo [ABCDEFGH] do exerc\u00edcio anterior e o plano IJK paralelo a AD.<\/p>\n<ol>\n<li>Determine as dimens\u00f5es da sec\u00e7\u00e3o [IJKL], supondo que I e J s\u00e3o pontos m\u00e9dios das arestas [EF] e [AE].<\/li>\n<li>Sendo $\\overline {EJ}\u00a0 = \\overline {EI} $, determine $\\overline {EJ} $ de modo que a sec\u00e7\u00e3o [IJKL] seja um quadrado.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_10861' onClick='GTTabs_show(1,10861)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_10861'>\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\/2012\/10\/Cubo-pag63-2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"10863\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10863\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-2.png\" data-orig-size=\"556,537\" 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-pag63-2.png\" class=\"alignright  wp-image-10863\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-2.png\" alt=\"\" width=\"334\" height=\"322\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-2.png 556w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-2-300x289.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-2-150x144.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/Cubo-pag63-2-400x386.png 400w\" sizes=\"auto, (max-width: 334px) 100vw, 334px\" \/><\/a>Se I e J s\u00e3o os pontos m\u00e9dios das arestas [EF] e [AE], a sec\u00e7\u00e3o produzida no cubo pelo plano IJK paralelo a AD \u00e9 um ret\u00e2ngulo com $\\overline {JK}\u00a0 = \\overline {IL}\u00a0 = 4$ cm e $\\overline {IJ}\u00a0 = \\overline {KL}\u00a0 = \\sqrt {{2^2} + {2^2}}\u00a0 = \\sqrt 8\u00a0 = 2\\sqrt 2 $ cm.<br \/>\n\u00ad<\/li>\n<li>Seja $\\overline {EJ}\u00a0 = \\overline {EI}\u00a0 = x$, com $0 &lt; x &lt; 4$, em cent\u00edmetros.\n<p>Para que o quadril\u00e1tero [IJKL] seja um quadrado, ter\u00e1 de ser: $\\overline {IJ}\u00a0 = \\overline {JK}\u00a0 = 4$.<\/p>\n<p>Assim, aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [IEJ], temos:<\/p>\n<p>$\\begin{array}{*{20}{l}} \u00a0 {\\begin{array}{*{20}{c}} \u00a0 {\\sqrt {{x^2} + {x^2}}\u00a0 = 4}&amp; \\wedge &amp;{0 &lt; x &lt; 4} \\end{array}}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}} \u00a0 {2{x^2} = 16}&amp; \\wedge &amp;{0 &lt; x &lt; 4} \\end{array}} \\\\ \u00a0 {}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}} \u00a0 {{x^2} = 8}&amp; \\wedge &amp;{0 &lt; x &lt; 4} \\end{array}} \\\\ \u00a0 {}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}} \u00a0 {x =\u00a0 \\mp \\sqrt 8 }&amp; \\wedge &amp;{0 &lt; x &lt; 4} \\end{array}} \\\\ \u00a0 {}&amp; \\Leftrightarrow &amp;{x = 2\\sqrt 2 } \\end{array}$<\/p>\n<p>Portanto, ter\u00e1 de ser $\\overline {EJ}\u00a0 = \\overline {EI}\u00a0 = 2\\sqrt 2 $ cm (o presente comprimento de [IJ]).<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_10861' onClick='GTTabs_show(0,10861)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere, ainda, o cubo [ABCDEFGH] do exerc\u00edcio anterior e o plano IJK paralelo a AD. Determine as dimens\u00f5es da sec\u00e7\u00e3o [IJKL], supondo que I e J s\u00e3o pontos m\u00e9dios das arestas&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20756,"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-10861","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":3104,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/10V1Pag063-2_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10861","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=10861"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10861\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20756"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10861"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10861"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10861"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=10861"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}