{"id":23389,"date":"2022-11-25T17:54:09","date_gmt":"2022-11-25T17:54:09","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=23389"},"modified":"2022-12-06T08:21:03","modified_gmt":"2022-12-06T08:21:03","slug":"um-quadrado-e-um-triangulo-equilatero","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=23389","title":{"rendered":"Um quadrado e um tri\u00e2ngulo equil\u00e1tero"},"content":{"rendered":"\n<p><ul id='GTTabs_ul_23389' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_23389' class='GTTabs_curr'><a  id=\"23389_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_23389' ><a  id=\"23389_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_23389'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>O pol\u00edgono [ABC] \u00e9 um tri\u00e2ngulo equil\u00e1tero e o pol\u00edgono [EFGH] \u00e9 um quadrado.<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23390\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23390\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6.png\" data-orig-size=\"677,217\" 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_Pag061-6\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6.png\" class=\"aligncenter wp-image-23390\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6-300x96.png\" alt=\"\" width=\"480\" height=\"154\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6-300x96.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6.png 677w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/a><\/p>\n<p>Quais s\u00e3o as abcissas dos pontos P e X?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_23389' onClick='GTTabs_show(1,23389)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_23389'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>O pol\u00edgono [ABC] \u00e9 um tri\u00e2ngulo equil\u00e1tero e o pol\u00edgono [EFGH] \u00e9 um quadrado.<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23390\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23390\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6.png\" data-orig-size=\"677,217\" 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_Pag061-6\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6.png\" class=\"aligncenter wp-image-23390\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6-300x96.png\" alt=\"\" width=\"480\" height=\"154\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6-300x96.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6.png 677w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/a><\/p>\n<p>Quais s\u00e3o as abcissas dos pontos P e X?<\/p>\n<\/blockquote>\n<p>\u00a0<\/p>\n<p>Aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [EGH], temos:<\/p>\n<p>\\[\\overline {EG} = \\sqrt {{{\\overline {EH} }^2} + {{\\overline {GH} }^2}} = \\sqrt {{3^2} + {3^2}} = \\sqrt {{3^2} \\times 2} = \\sqrt {{3^2}} \\times \\sqrt 2 = 3\\sqrt 2 \\]<\/p>\n<p>Logo, \\(P \\to 3\\sqrt 2 \\).<\/p>\n<p>\u00a0<\/p>\n<p>Seja \\(M \\to \\frac{7}{2}\\), o ponto m\u00e9dio do segmento de reta [AB], base do tri\u00e2ngulo equil\u00e1tero.<\/p>\n<p>Determinemos a altura do tri\u00e2ngulo, por aplica\u00e7\u00e3o do Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [BCM]:<\/p>\n<p>\\[\\overline {CM} = \\sqrt {{{\\overline {BC} }^2} &#8211; {{\\overline {BM} }^2}} = \\sqrt {{3^2} &#8211; {{\\left( {\\frac{3}{2}} \\right)}^2}} = \\sqrt {{3^2} &#8211; \\frac{{{3^2}}}{4}} = \\sqrt {\\frac{{4 \\times {3^2}}}{4} &#8211; \\frac{{{3^2}}}{4}} = \\sqrt {\\frac{{3 \\times {3^2}}}{4}} = \\sqrt {\\frac{{{3^2}}}{4}} \\times \\sqrt 3 = \\frac{3}{2}\\sqrt 3 \\]<\/p>\n<p>Logo, \\(X \\to \\frac{{7 + 3\\sqrt 3 }}{2}\\).<\/p>\n<p>\u00a0<\/p>\n<h6>Os c\u00e1lculos sem simplifica\u00e7\u00e3o dos radicais<\/h6>\n<ul style=\"list-style-type: square;\">\n<li>\\(\\overline {EG} = \\sqrt {{{\\overline {EH} }^2} + {{\\overline {GH} }^2}} = \\sqrt {{3^2} + {3^2}} = \\sqrt {9 + 9} = \\sqrt {18} \\)<br \/>Logo, \\(P \\to \\sqrt {18} \\).<br \/><br \/><\/li>\n<li>\\(\\overline {CM} = \\sqrt {{{\\overline {BC} }^2} &#8211; {{\\overline {BM} }^2}} = \\sqrt {{3^2} &#8211; {{\\left( {\\frac{3}{2}} \\right)}^2}} = \\sqrt {9 &#8211; \\frac{9}{4}} = \\sqrt {\\frac{{36}}{4} &#8211; \\frac{9}{4}} = \\sqrt {\\frac{{27}}{4}} = \\sqrt {6,75} \\)<br \/>Logo, \\(X \\to 3,5 + \\sqrt {6,75} \\).<\/li>\n<\/ul>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_23389' onClick='GTTabs_show(0,23389)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado O pol\u00edgono [ABC] \u00e9 um tri\u00e2ngulo equil\u00e1tero e o pol\u00edgono [EFGH] \u00e9 um quadrado. Quais s\u00e3o as abcissas dos pontos P e X? Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":23391,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,682],"tags":[424,67,118],"series":[],"class_list":["post-23389","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-teorema-de-pitagoras"],"views":320,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag061-6_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/23389","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=23389"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/23389\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/23391"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=23389"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=23389"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=23389"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=23389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}