{"id":4349,"date":"2010-10-20T19:04:22","date_gmt":"2010-10-20T18:04:22","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=4349"},"modified":"2022-01-21T02:25:36","modified_gmt":"2022-01-21T02:25:36","slug":"um-triangulo-equilatero","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=4349","title":{"rendered":"Um tri\u00e2ngulo equil\u00e1tero"},"content":{"rendered":"<p><ul id='GTTabs_ul_4349' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_4349' class='GTTabs_curr'><a  id=\"4349_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_4349' ><a  id=\"4349_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_4349'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"4350\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=4350\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57.jpg\" data-orig-size=\"372,321\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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;}\" data-image-title=\"Tri\u00e2ngulo equil\u00e1tero\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57.jpg\" class=\"alignright wp-image-4350\" title=\"Tri\u00e2ngulo equil\u00e1tero\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57-300x258.jpg\" alt=\"\" width=\"300\" height=\"259\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57-300x258.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57-150x129.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57.jpg 372w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Observe a figura onde est\u00e1 representado um tri\u00e2ngulo equil\u00e1tero inscrito numa circunfer\u00eancia de raio 6 unidades. O ponto C pertence ao eixo das ordenadas.<\/p>\n<ol>\n<li>Indique as coordenadas dos v\u00e9rtices do tri\u00e2ngulo.<\/li>\n<li>Indique as coordenadas do ortocentro do tri\u00e2ngulo (ponto de intersec\u00e7\u00e3o das alturas do tri\u00e2ngulo).<\/li>\n<li>Se o tri\u00e2ngulo rodar 90\u00ba em torno de O, quais ser\u00e3o agora as coordenadas dos seus v\u00e9rtices?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_4349' onClick='GTTabs_show(1,4349)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_4349'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"4350\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=4350\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57.jpg\" data-orig-size=\"372,321\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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;}\" data-image-title=\"Tri\u00e2ngulo equil\u00e1tero\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57.jpg\" class=\"alignright wp-image-4350\" title=\"Tri\u00e2ngulo equil\u00e1tero\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57-300x258.jpg\" alt=\"\" width=\"300\" height=\"259\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57-300x258.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57-150x129.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag97-57.jpg 372w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Como sabemos, no caso do raio da circunfer\u00eancia ser 1 unidade, as coordenadas dos v\u00e9rtices do tri\u00e2ngulo seriam traduzidas pelo cosseno e pelo seno, respetivamente, dos \u00e2ngulos generalizados com lados extremidades contendo cada um desses v\u00e9rtices.<br \/>\nComo o raio da circunfer\u00eancia \u00e9 6 unidades, ent\u00e3o essas coordenadas vir\u00e3o multiplicadas por 6.<\/p>\n<p>Assim, temos:<br \/>\n&#8211; Coordenadas do ponto A:<br \/>\n$(6\\cos 210{}^\\text{o},6\\,sen\\,210{}^\\text{o})=(-6\\times \\frac{\\sqrt{3}}{2},-6\\times \\frac{1}{2})=(-3\\sqrt{3},-3)$<br \/>\n&#8211; Coordenadas do ponto B:<br \/>\n$(6\\cos 330{}^\\text{o},6\\,sen\\,330{}^\\text{o})=(6\\times \\frac{\\sqrt{3}}{2},-6\\times \\frac{1}{2})=(3\\sqrt{3},-3)$<br \/>\n&#8211; Coordenadas do ponto C:<br \/>\n$(6\\cos 90{}^\\text{o},6\\,sen\\,90{}^\\text{o})=(6\\times 0,6\\times 1)=(0,6)$<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Como sabemos,\u00a0o segmento altura do tri\u00e2ngulo equil\u00e1tero \u00e9 perpendicular \u00e0 base, no seu ponto m\u00e9dio.<br \/>\nPor outro lado, sabemos que a mediatriz de qualquer corda de uma circunfer\u00eancia cont\u00e9m o seu centro.<br \/>\nComo os lados do tri\u00e2ngulo s\u00e3o cordas da mesma circunfer\u00eancia, ent\u00e3o as suas mediatrizes cont\u00eam as tr\u00eas alturas do tri\u00e2ngulo, que se intersectam no seu centro. Logo, o ortocentro do tri\u00e2ngulo \u00e9 O (0,0).<br \/>\n\u00ad<\/li>\n<li>Se o tri\u00e2ngulo rodar 90\u00ba em torno de O, as coordenadas dos seus v\u00e9rtices ser\u00e3o:<br \/>\n&#8211; A&#8217;: $6\\cos 300{}^\\text{o},6\\,sen\\,300{}^\\text{o})=(6\\times \\frac{1}{2},-6\\times \\frac{\\sqrt{3}}{2})=(3,-3\\sqrt{3})$<br \/>\n&#8211; B&#8217;: $(6\\cos 420{}^\\text{o},6\\,sen\\,420{}^\\text{o})=(6\\times \\frac{1}{2},6\\times \\frac{\\sqrt{3}}{2})=(3,3\\sqrt{3})$<br \/>\n&#8211; C&#8217;: $(6\\cos 180{}^\\text{o},6\\,sen\\,180{}^\\text{o})=(-6\\times 1,-6\\times 0)=(-6,0)$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_4349' onClick='GTTabs_show(0,4349)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Observe a figura onde est\u00e1 representado um tri\u00e2ngulo equil\u00e1tero inscrito numa circunfer\u00eancia de raio 6 unidades. O ponto C pertence ao eixo das ordenadas. Indique as coordenadas dos v\u00e9rtices do tri\u00e2ngulo.&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20806,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,99],"tags":[422,423],"series":[],"class_list":["post-4349","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-trigonometria","tag-11-o-ano","tag-trigonometria"],"views":3037,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/11V1Pag097-57_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/4349","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=4349"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/4349\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20806"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4349"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4349"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4349"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=4349"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}