{"id":14204,"date":"2018-03-30T22:51:59","date_gmt":"2018-03-30T21:51:59","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14204"},"modified":"2022-01-11T11:38:41","modified_gmt":"2022-01-11T11:38:41","slug":"o-perimetro-do-triangulo-abc","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14204","title":{"rendered":"O per\u00edmetro do tri\u00e2ngulo [ABC]"},"content":{"rendered":"<p><ul id='GTTabs_ul_14204' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14204' class='GTTabs_curr'><a  id=\"14204_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14204' ><a  id=\"14204_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_14204'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-7.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14205\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14205\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-7.png\" data-orig-size=\"242,374\" 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\/9V2Pag60-7.png\" class=\"alignright size-medium wp-image-14205\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-7-194x300.png\" alt=\"\" width=\"194\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-7-194x300.png 194w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-7.png 242w\" sizes=\"auto, (max-width: 194px) 100vw, 194px\" \/><\/a>Na figura est\u00e1 representado um tri\u00e2ngulo ret\u00e2ngulo em <em>B<\/em> e um ponto <em>D<\/em> no lado [<em>BC<\/em>] tal que \\(B\\widehat AD = B\\widehat CA = 30^\\circ \\).<\/p>\n<p>Sabendo que \\(\\overline {AD} = 4\\) cm, determina o valor exato do\u00a0per\u00edmetro do tri\u00e2ngulo [<em>ABC<\/em>].<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14204' onClick='GTTabs_show(1,14204)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14204'>\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\/9V2Pag60-7.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14205\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14205\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-7.png\" data-orig-size=\"242,374\" 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\/9V2Pag60-7.png\" class=\"alignright size-medium wp-image-14205\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-7-194x300.png\" alt=\"\" width=\"194\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-7-194x300.png 194w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-7.png 242w\" sizes=\"auto, (max-width: 194px) 100vw, 194px\" \/><\/a>No tri\u00e2ngulo ret\u00e2ngulo [<em>ABD<\/em>], vem \\(\\cos B\\widehat AD = \\frac{{\\overline {AB} }}{{\\overline {AD} }}\\), donde:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{\\cos 30^\\circ = \\frac{{\\overline {AB} }}{4}}&amp; \\Leftrightarrow &amp;{\\frac{{\\sqrt 3 }}{2} = \\frac{{\\overline {AB} }}{4}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {AB} = 2\\sqrt 3 }\\end{array}\\]<\/p>\n<p>No tri\u00e2ngulo ret\u00e2ngulo [<em>ABC<\/em>], vem \\({\\mathop{\\rm sen}\\nolimits} A\\widehat CB = \\frac{{\\overline {AB} }}{{\\overline {AC} }}\\) e \\({\\mathop{\\rm tg}\\nolimits} A\\widehat CB = \\frac{{\\overline {AB} }}{{\\overline {BC} }}\\), donde temos, respetivamente:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{\\mathop{\\rm sen}\\nolimits} 30^\\circ = \\frac{{2\\sqrt 3 }}{{\\overline {AC} }}}&amp; \\Leftrightarrow &amp;{\\frac{1}{2} = \\frac{{2\\sqrt 3 }}{{\\overline {AC} }}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {AC} = 4\\sqrt 3 }\\end{array}\\]<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{\\mathop{\\rm tg}\\nolimits} 30^\\circ = \\frac{{2\\sqrt 3 }}{{\\overline {BC} }}}&amp; \\Leftrightarrow &amp;{\\frac{{\\sqrt 3 }}{3} = \\frac{{2\\sqrt 3 }}{{\\overline {BC} }}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {BC} = 6}\\end{array}\\]<\/p>\n<p>Logo, em cent\u00edmetros, o per\u00edmetro do tri\u00e2ngulo [<em>ABC<\/em>] \u00e9:<\/p>\n<p>\\[{P_{\\left[ {ABC} \\right]}} = \\overline {AB} + \\overline {BC} + \\overline {AC} = 2\\sqrt 3 + 6 + 4\\sqrt 3 = 6 + 6\\sqrt 3 \\]<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14204' onClick='GTTabs_show(0,14204)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e1 representado um tri\u00e2ngulo ret\u00e2ngulo em B e um ponto D no lado [BC] tal que \\(B\\widehat AD = B\\widehat CA = 30^\\circ \\). Sabendo que \\(\\overline {AD} =&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14207,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,489],"tags":[426,491,493,490,492],"series":[],"class_list":["post-14204","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","tag-tangente"],"views":1231,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag60-7a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14204","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=14204"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14204\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14207"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14204"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14204"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14204"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14204"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}