{"id":3860,"date":"2010-10-09T02:09:54","date_gmt":"2010-10-09T01:09:54","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=3860"},"modified":"2022-01-21T01:49:23","modified_gmt":"2022-01-21T01:49:23","slug":"um-hexagono-regular","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=3860","title":{"rendered":"Um hex\u00e1gono regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_3860' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_3860' class='GTTabs_curr'><a  id=\"3860_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_3860' ><a  id=\"3860_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_3860'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"3863\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=3863\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44.jpg\" data-orig-size=\"333,384\" 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=\"Hex\u00e1gono regular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44.jpg\" class=\"alignright wp-image-3863\" title=\"Hex\u00e1gono regular\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44-260x300.jpg\" alt=\"\" width=\"240\" height=\"277\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44-260x300.jpg 260w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44-130x150.jpg 130w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44.jpg 333w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Num referencial cartesiano, est\u00e1 representada uma circunfer\u00eancia com raio de uma unidade de comprimento e um hex\u00e1gono [CDEFGH].<\/p>\n<ol>\n<li>Explique porque \u00e9 que sabemos que a abcissa de D \u00e9 $\\cos \\frac{\\pi }{3}$.<\/li>\n<li>Determine as coordenadas exactas dos pontos E e G.<\/li>\n<li>Qual a medida do comprimento do arco CE na unidade considerada?<\/li>\n<li>Se o raio passasse a ter 5 unidades de comprimento, qual era a abcissa de D?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_3860' onClick='GTTabs_show(1,3860)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_3860'>\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\/44.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"3863\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=3863\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44.jpg\" data-orig-size=\"333,384\" 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=\"Hex\u00e1gono regular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44.jpg\" class=\"alignright wp-image-3863\" title=\"Hex\u00e1gono regular\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44-260x300.jpg\" alt=\"\" width=\"240\" height=\"277\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44-260x300.jpg 260w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44-130x150.jpg 130w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/44.jpg 333w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Como se sabe, $C\\widehat{O}D=\\frac{360{}^\\text{o}}{6}=60{}^\\text{o}=\\frac{\\pi }{3}rad$.<br \/>\nComo a circunfer\u00eancia tem de raio uma unidade de comprimento, ent\u00e3o $D\\,(\\cos C\\widehat{O}D,sen\\,C\\widehat{O}D)=(\\cos \\frac{\\pi }{3},sen\\,\\frac{\\pi }{3})$.<br \/>\nLogo, a abcissa de D \u00e9 $\\cos \\frac{\\pi }{3}$.<br \/>\n\u00ad<\/li>\n<li>$E\\,(\\cos \\frac{2\\pi }{3},sen\\,\\frac{2\\pi }{3})=(-\\cos \\frac{\\pi }{3},sen\\,\\frac{\\pi }{3})=(-\\frac{1}{2},\\frac{\\sqrt{3}}{2})$.<br \/>\n$G\\,(\\cos \\frac{4\\pi }{3},sen\\,\\frac{4\\pi }{3})=(-\\cos \\frac{\\pi }{3},-sen\\,\\frac{\\pi }{3})=(-\\frac{1}{2},-\\frac{\\sqrt{3}}{2})$.<br \/>\n\u00ad<\/li>\n<li>Designando o comprimento do arco CE por <em>x<\/em> e tendo em considera\u00e7\u00e3o a defini\u00e7\u00e3o de radiano, vem: \\[\\frac{1\\,(rad)}{1\\,(u.c.)}=\\frac{\\frac{2\\pi }{3}(rad)}{x\\,(u.c.)}\\Leftrightarrow x=\\frac{2\\pi }{3}(u.c.)\\]<br \/>\nPortanto, arco CE tem de comprimento\u00a0$\\frac{2\\pi }{3}$ unidades.<br \/>\n\u00ad<\/li>\n<li>Se o raio da circunfer\u00eancia que circunscreve o hex\u00e1gono passasse a ter 5 unidades de comprimento, a semi-recta $\\dot{O}D$ intersectaria a circunfer\u00eancia conc\u00eantrica de raio\u00a0unit\u00e1rio no ponto $D&#8217;\\,(\\cos \\frac{\\pi }{3},sen\\,\\frac{\\pi }{3})=(\\frac{1}{2},\\frac{\\sqrt{3}}{2})$.<br \/>\nNessa circunst\u00e2ncia, seria $\\overrightarrow{OD}=5\\times \\overrightarrow{OD&#8217;}$ e,\u00a0consequentemente, $D\\,(5\\times \\cos \\frac{\\pi }{3},5\\times sen\\,\\frac{\\pi }{3})=(\\frac{5}{2},\\frac{5\\sqrt{3}}{2})$.<br \/>\nPortanto, a abcissa de D seria 2,5.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_3860' onClick='GTTabs_show(0,3860)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Num referencial cartesiano, est\u00e1 representada uma circunfer\u00eancia com raio de uma unidade de comprimento e um hex\u00e1gono [CDEFGH]. Explique porque \u00e9 que sabemos que a abcissa de D \u00e9 $\\cos \\frac{\\pi&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20799,"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-3860","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":2283,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/11V1Pag094-44-b_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/3860","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=3860"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/3860\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20799"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3860"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3860"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3860"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=3860"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}