{"id":14196,"date":"2018-03-30T21:45:11","date_gmt":"2018-03-30T20:45:11","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14196"},"modified":"2022-01-11T11:35:45","modified_gmt":"2022-01-11T11:35:45","slug":"a-area-do-triangulo-amc","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14196","title":{"rendered":"A \u00e1rea do tri\u00e2ngulo [AMC]"},"content":{"rendered":"<p><ul id='GTTabs_ul_14196' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14196' class='GTTabs_curr'><a  id=\"14196_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14196' ><a  id=\"14196_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_14196'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14197\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14197\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6.png\" data-orig-size=\"249,363\" 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=\"Tri\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6.png\" class=\"alignright size-medium wp-image-14197\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6-206x300.png\" alt=\"\" width=\"206\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6-206x300.png 206w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6.png 249w\" sizes=\"auto, (max-width: 206px) 100vw, 206px\" \/><\/a>Considera um tri\u00e2ngulo [<em>ABC<\/em>], ret\u00e2ngulo em <em>B<\/em>, e o ponto m\u00e9dio <em>M<\/em> de [<em>AB<\/em>].<\/p>\n<p>Sabendo que\u00a0\\(B\\widehat AC = 60^\\circ \\) e que \\(\\overline {AC} = 6\\) cm, determina o valor exato da \u00e1rea do tri\u00e2ngulo [<em>AMC<\/em>]<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14196' onClick='GTTabs_show(1,14196)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14196'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14197\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14197\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6.png\" data-orig-size=\"249,363\" 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=\"Tri\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6.png\" class=\"alignright size-medium wp-image-14197\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6-206x300.png\" alt=\"\" width=\"206\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6-206x300.png 206w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6.png 249w\" sizes=\"auto, (max-width: 206px) 100vw, 206px\" \/><\/a>No tri\u00e2ngulo ret\u00e2ngulo [<em>ABC<\/em>], vem \\(\\cos B\\widehat AC = \\frac{{\\overline {AB} }}{{\\overline {AC} }}\\), donde:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{\\cos 60^\\circ = \\frac{{\\overline {AB} }}{6}}&amp; \\Leftrightarrow &amp;{\\frac{1}{2} = \\frac{{\\overline {AB} }}{6}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {AB} = 3}\\end{array}\\]<\/p>\n<p>Ainda no tri\u00e2ngulo [<em>ABC<\/em>], vem \\({\\mathop{\\rm sen}\\nolimits} B\\widehat AC = \\frac{{\\overline {BC} }}{{\\overline {AC} }}\\), donde:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{\\mathop{\\rm sen}\\nolimits} 60^\\circ = \\frac{{\\overline {BC} }}{6}}&amp; \\Leftrightarrow &amp;{\\frac{{\\sqrt 3 }}{2} = \\frac{{\\overline {BC} }}{6}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {BC} = 3\\sqrt 3 }\\end{array}\\]<\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Em alternativa<\/strong><\/span>, vem por aplica\u00e7\u00e3o do Teorema de Pit\u00e1goras:\u00a0\\(\\overline {BC} = \\sqrt {{{\\overline {AC} }^2} &#8211; {{\\overline {AB} }^2}} = \\sqrt {{6^2} &#8211; {3^2}} = \\sqrt {27} = 3\\sqrt 3 \\)<br \/>\n\u00ad<\/p>\n<p>Logo, em cent\u00edmetros quadrados, a \u00e1rea do tri\u00e2ngulo [<em>AMC<\/em>] \u00e9:<\/p>\n<p>\\[{A_{\\left[ {AMC} \\right]}} = \\frac{{\\overline {AM} \\times \\overline {BC} }}{2} = \\frac{{1,5 \\times 3\\sqrt 3 }}{2} = \\frac{{9\\sqrt 3 }}{4}\\]<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14196' onClick='GTTabs_show(0,14196)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considera um tri\u00e2ngulo [ABC], ret\u00e2ngulo em B, e o ponto m\u00e9dio M de [AB]. Sabendo que\u00a0\\(B\\widehat AC = 60^\\circ \\) e que \\(\\overline {AC} = 6\\) cm, determina o valor exato&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14198,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,489],"tags":[426,491,493,490],"series":[],"class_list":["post-14196","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-trigonometria-9--ano","tag-9-o-ano","tag-cosseno","tag-razao-trigonometrica","tag-seno"],"views":2970,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-6a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14196","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=14196"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14196\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14198"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14196"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14196"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14196"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14196"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}