{"id":23173,"date":"2022-11-19T14:22:27","date_gmt":"2022-11-19T14:22:27","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=23173"},"modified":"2022-11-24T19:03:08","modified_gmt":"2022-11-24T19:03:08","slug":"a-janela","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=23173","title":{"rendered":"A janela"},"content":{"rendered":"\n<p><ul id='GTTabs_ul_23173' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_23173' class='GTTabs_curr'><a  id=\"23173_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_23173' ><a  id=\"23173_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_23173'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23174\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23174\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-1.png\" data-orig-size=\"360,245\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"8_Pag055-T7-1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-1.png\" class=\"alignright wp-image-23174 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-1-300x204.png\" alt=\"\" width=\"300\" height=\"204\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-1-300x204.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-1.png 360w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Na figura ao lado, est\u00e1 a fotografia de uma janela. No gradeamento exterior, podem observar-se diferentes pol\u00edgonos, entre os quais v\u00e1rios ret\u00e2ngulos e dois quadrados com o mesmo centro (os v\u00e9rtices do quadrado mais pequeno s\u00e3o os pontos m\u00e9dios das semidiagonais do quadrado maior).<\/p>\n<p>Observa o seguinte esquema do gradeamento da janela.<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23178\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23178\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2.png\" data-orig-size=\"500,264\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"8_Pag055-T7-2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2.png\" class=\"aligncenter wp-image-23178 size-full\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2.png\" alt=\"\" width=\"500\" height=\"264\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2.png 500w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-300x158.png 300w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/p>\n<p>Se o ferro para construir este tipo de gradeamento se vender em barras de 3 metros de comprimento, qual \u00e9 o n\u00famero m\u00ednimo de barras necess\u00e1rias para construir o gradeamento desta janela?<br \/>Apresenta todos os c\u00e1lculos que efetuares.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_23173' onClick='GTTabs_show(1,23173)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_23173'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><\/p>\n<blockquote>\n<p>Observa o seguinte esquema do gradeamento da janela.<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-_Nome-pontos.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23180\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23180\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-_Nome-pontos.png\" data-orig-size=\"500,264\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"8_Pag055-T7-2 _Nome-pontos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-_Nome-pontos.png\" class=\"aligncenter wp-image-23180 size-full\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-_Nome-pontos.png\" alt=\"\" width=\"500\" height=\"264\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-_Nome-pontos.png 500w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-_Nome-pontos-300x158.png 300w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/p>\n<\/blockquote>\n<blockquote>\n<p>Se o ferro para construir este tipo de gradeamento se vender em barras de 3 metros de comprimento, qual \u00e9 o n\u00famero m\u00ednimo de barras necess\u00e1rias para construir o gradeamento desta janela?<br \/>Apresenta todos os c\u00e1lculos que efetuares.<\/p>\n<\/blockquote>\n<p>\u00a0<\/p>\n<p>Comprimento total do gradeamento da janela, em metros:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{C_T}}&amp; = &amp;{8 \\times 0,7 + 8 \\times 0,2 + 2 \\times 0,1 + 4 \\times 0,35 + 2 \\times \\sqrt {{{0,7}^2} + {{0,7}^2}} }\\\\{}&amp; = &amp;{5,6 + 1,6 + 0,2 + 1,4 + 2\\sqrt {0,49 + 0,49} }\\\\{}&amp; = &amp;{8,8 + 2 \\times \\sqrt {0,98} }\\end{array}\\]<\/p>\n<p>Como \\(\\frac{{{C_T}}}{3} = \\frac{{8,8 + 2 \\times \\sqrt {0,98} }}{3} \\approx 3,59\\), ent\u00e3o s\u00e3o necess\u00e1rias, no m\u00ednimo, quatro barras de 3 metros de comprimento para construir o gradeamento desta janela.<\/p>\n<p>\u00a0<\/p>\n<h6>C\u00e1lculos mais detalhados<\/h6>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-_Nome-pontos.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23180\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23180\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-_Nome-pontos.png\" data-orig-size=\"500,264\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"8_Pag055-T7-2 _Nome-pontos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-_Nome-pontos.png\" class=\"aligncenter wp-image-23180 size-full\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-_Nome-pontos.png\" alt=\"\" width=\"500\" height=\"264\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-_Nome-pontos.png 500w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-2-_Nome-pontos-300x158.png 300w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/p>\n<p>Comprimento total das seis pe\u00e7as verticais, de 70 cm de comprimento:<\/p>\n<ul>\n<li>\\({C_V} = \\overline {AL} + \\overline {BK} + \\overline {MJ} + \\overline {NI} + \\overline {EH} + \\overline {FG} = 6 \\times 0,7 = 4,2\\) m<\/li>\n<\/ul>\n<p>Comprimento total das duas bases de apoio no parapeito da janela:<\/p>\n<ul>\n<li>\\({C_B} = \\overline {CM} + \\overline {DN} = 2 \\times 0,1 = 0,2\\) m<\/li>\n<\/ul>\n<p>Comprimento total das duas pe\u00e7as horizontais mais exteriores:<\/p>\n<ul>\n<li>\\({C_H} = \\overline {LG} + \\overline {AF} = 2 \\times \\left( {0,2 + 0,2 + 0,7 + 0,2 + 0,2} \\right) = 3,0\\) m<\/li>\n<\/ul>\n<p>Comprimento da pe\u00e7a diagonal do quadrado grande:<\/p>\n<ul>\n<li>\\({C_D} = \\sqrt {{{\\overline {CJ} }^2} + {{\\overline {CD} }^2}} = \\sqrt {{{0,7}^2} + {{0,7}^2}} = \\sqrt {0,49 + 0,49} = \\sqrt {0,98} \\) m<\/li>\n<\/ul>\n<p>Comprimento da pe\u00e7a do lado do quadrado pequeno (ver a nota seguinte):<\/p>\n<ul>\n<li>\\({C_{LQP}} = \\frac{{\\overline {CD} }}{2} = \\frac{{0,7}}{2} = 0,35\\)<\/li>\n<\/ul>\n<p><strong>Nota<\/strong>:<br \/>Como os v\u00e9rtices do quadrado mais pequeno s\u00e3o os pontos m\u00e9dios das semidiagonais do quadrado maior, ent\u00e3o os tri\u00e2ngulos [MNS] e [ROS] s\u00e3o semelhantes, tendo-se: \\(\\frac{{\\overline {RO} }}{{\\overline {MN} }} = \\frac{{\\overline {RS} }}{{\\overline {MS} }} = \\frac{{\\overline {OS} }}{{\\overline {NS} }} = \\frac{1}{2}\\).<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_23173' onClick='GTTabs_show(0,23173)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura ao lado, est\u00e1 a fotografia de uma janela. No gradeamento exterior, podem observar-se diferentes pol\u00edgonos, entre os quais v\u00e1rios ret\u00e2ngulos e dois quadrados com o mesmo centro (os v\u00e9rtices&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":23176,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,682],"tags":[424,67,149,118],"series":[],"class_list":["post-23173","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-8--ano","category-aplicando","category-teorema-de-pitagoras","tag-8-o-ano","tag-geometria","tag-semelhanca-de-triangulos","tag-teorema-de-pitagoras"],"views":231,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag055-T7-1_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/23173","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=23173"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/23173\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/23176"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=23173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=23173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=23173"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=23173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}