{"id":10203,"date":"2012-10-11T16:53:16","date_gmt":"2012-10-11T15:53:16","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=10203"},"modified":"2022-01-20T22:49:41","modified_gmt":"2022-01-20T22:49:41","slug":"a-piramide-de-queops-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=10203","title":{"rendered":"A pir\u00e2mide de Qu\u00e9ops"},"content":{"rendered":"<p style=\"text-align: left;\"><ul id='GTTabs_ul_10203' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_10203' class='GTTabs_curr'><a  id=\"10203_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_10203' ><a  id=\"10203_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_10203'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"10206\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10206\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops.jpg\" data-orig-size=\"842,272\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349971128&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=\"Tri\u00e2ngulos semelhantes\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops.jpg\" class=\"alignright  wp-image-10206\" title=\"Tri\u00e2ngulos semelhantes\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops.jpg\" alt=\"\" width=\"505\" height=\"163\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops.jpg 842w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops-300x96.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops-150x48.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops-400x129.jpg 400w\" sizes=\"auto, (max-width: 505px) 100vw, 505px\" \/><\/a>Conta-se que <a href=\"http:\/\/en.wikipedia.org\/wiki\/Thales\" target=\"_blank\" rel=\"noopener\">Thales de Mileto<\/a> (s\u00e9c. VI a.C.), considerado por alguns autores como um dos sete s\u00e1bios da Antiguidade, se ofereceu para determinar a altura da <a href=\"http:\/\/en.wikipedia.org\/wiki\/Egyptian_pyramids\" target=\"_blank\" rel=\"noopener\">pir\u00e2mide de Qu\u00e9ops<\/a>, sem escalar o monumento.<\/p>\n<p>Segundo a lenda, a prova ter-se-\u00e1 realizado na presen\u00e7a do fara\u00f3 Amasis. Thales espetou perpendicularmente ao ch\u00e3o a sua bengala e mediu as sombras da bengala e da pir\u00e2mide. Ap\u00f3s os c\u00e1lculos r\u00e1pidos, Thales obteve a resposta desejada.<\/p>\n<p>Em que se baseou o racioc\u00ednio de Thales?<\/p>\n<p>Observe cuidadosamente as figuras, suponha que a aresta da base da pir\u00e2mide de Qu\u00e9ops tem $230$ m de comprimento, a sombra da pir\u00e2mide e da bengala s\u00e3o, respetivamente, $323$ m e $2,5$ m e que a bengala tem $80$ cm de comprimento. Qual a altura da pir\u00e2mide?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_10203' onClick='GTTabs_show(1,10203)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_10203'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p style=\"text-align: center;\">\u00a0<a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"10206\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10206\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops.jpg\" data-orig-size=\"842,272\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349971128&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=\"Tri\u00e2ngulos semelhantes\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops.jpg\" class=\"aligncenter  wp-image-10206\" title=\"Tri\u00e2ngulos semelhantes\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops.jpg\" alt=\"\" width=\"505\" height=\"163\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops.jpg 842w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops-300x96.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops-150x48.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/queops-400x129.jpg 400w\" sizes=\"auto, (max-width: 505px) 100vw, 505px\" \/><\/a><\/p>\n<\/p>\n<p>\u00c0 superf\u00edcie da Terra, os raios solares consideram-se paralelos. Desta forma, os \u00e2ngulos ACB e DFE s\u00e3o geometricamente iguais.<\/p>\n<p>Como os tri\u00e2ngulos [ABC] e [DEF] s\u00e3o ambos ret\u00e2ngulos e possuem um \u00e2ngulo agudo igual, ent\u00e3o s\u00e3o semelhantes.<\/p>\n<p>Portanto, o racioc\u00ednio de Thales baseou-se na semelhan\u00e7a de tri\u00e2ngulos: $\\frac{{\\overline {AB} }}{{\\overline {DE} }} = \\frac{{\\overline {BC} }}{{\\overline {EF} }} = \\frac{{\\overline {AC} }}{{\\overline {DF} }}$.<\/p>\n<p>Considerando a igualdade entre as duas primeiras fra\u00e7\u00f5es, temos:<\/p>\n<p>$$\\begin{array}{*{20}{l}} \u00a0 {\\frac{{\\overline {AB} }}{{0,8}} = \\frac{{\\frac{{230}}{2} + 323}}{{2,5}}}&amp; \\Leftrightarrow &amp;{\\overline {AB}\u00a0 = \\frac{{438 \\times 0,8}}{{2,5}}} \\\\ \u00a0 {}&amp; \\Leftrightarrow &amp;{\\overline {AB}\u00a0 = 140,16} \\end{array}$$<\/p>\n<p>A pir\u00e2mide de Qu\u00e9ops tem $140,16$ m de altura.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_10203' onClick='GTTabs_show(0,10203)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Conta-se que Thales de Mileto (s\u00e9c. VI a.C.), considerado por alguns autores como um dos sete s\u00e1bios da Antiguidade, se ofereceu para determinar a altura da pir\u00e2mide de Qu\u00e9ops, sem escalar&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20789,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,322],"tags":[429,67,430,149],"series":[],"class_list":["post-10203","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-10-o-ano","category-aplicando","category-modulo-inicial","tag-10-o-ano","tag-geometria","tag-modulo-inicial","tag-semelhanca-de-triangulos"],"views":7336,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/10V1Pag039-30_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10203","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=10203"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10203\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20789"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10203"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10203"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10203"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=10203"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}