{"id":13453,"date":"2018-02-03T16:55:04","date_gmt":"2018-02-03T16:55:04","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=13453"},"modified":"2022-01-17T16:03:15","modified_gmt":"2022-01-17T16:03:15","slug":"um-hexagono-regular-inscrito-numa-circunferencia","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=13453","title":{"rendered":"Um hex\u00e1gono regular inscrito numa circunfer\u00eancia"},"content":{"rendered":"<p><ul id='GTTabs_ul_13453' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_13453' class='GTTabs_curr'><a  id=\"13453_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_13453' ><a  id=\"13453_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_13453'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Desenha um hex\u00e1gono regular inscrito numa circunfer\u00eancia e os raios correspondentes aos extremos de cada um dos lados do hex\u00e1gono.<\/p>\n<ol>\n<li>Classifica cada um dos tri\u00e2ngulos obtidos.<\/li>\n<li>Que rela\u00e7\u00e3o existe entre o comprimento do lado do hex\u00e1gono e o comprimento do raio da circunfer\u00eancia circunscrita ao hex\u00e1gono?<\/li>\n<li>Se a circunfer\u00eancia tiver 5 cm de raio, qual \u00e9 a \u00e1rea do hex\u00e1gono nela inscrito?<br \/>\nApresenta todos os c\u00e1lculos que efetuares e o valor arredondado \u00e0s unidades.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_13453' onClick='GTTabs_show(1,13453)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_13453'>\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\/2018\/02\/2017-18-MF9P1-pag145-16.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13454\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13454\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/2017-18-MF9P1-pag145-16.png\" data-orig-size=\"439,406\" 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=\"Hex\u00e1gono regular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/2017-18-MF9P1-pag145-16.png\" class=\"alignright wp-image-13454\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/2017-18-MF9P1-pag145-16-300x277.png\" alt=\"\" width=\"340\" height=\"314\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/2017-18-MF9P1-pag145-16-300x277.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/2017-18-MF9P1-pag145-16.png 439w\" sizes=\"auto, (max-width: 340px) 100vw, 340px\" \/><\/a>Os seis tri\u00e2ngulos obtidos s\u00e3o equil\u00e1teros e geometricamente iguais.<br \/>\nComo a circunfer\u00eancia foi dividida em seis arcos geometricamente iguais, ent\u00e3o os seis \u00e2ngulos ao centro s\u00e3o geometricamente iguais e com amplitude de 60 graus.<br \/>\nConsidere-se agora, por exemplo, o tri\u00e2ngulo [OAB].<br \/>\nDois dos lados deste tri\u00e2ngulo s\u00e3o raios da circunfer\u00eancia, os lados [OA] e [OB]. Como, num tri\u00e2ngulo, a lados iguais se op\u00f5em \u00e2ngulos iguais, ent\u00e3o\u00a0\\(O\\widehat AB = O\\widehat BA = \\frac{{180^\\circ &#8211; A\\widehat OB}}{2} = \\frac{{180^\\circ &#8211; 60^\\circ }}{2} = 60^\\circ \\).<br \/>\nOra, sendo equi\u00e2ngulo, o tri\u00e2ngulo [OAB] \u00e9 tamb\u00e9m equil\u00e1tero, c.q.m.<br \/>\n\u00ad<\/li>\n<li>Da prova anterior, resulta que s\u00e3o iguais o comprimento do lado do hex\u00e1gono e o comprimento do raio da circunfer\u00eancia circunscrita ao hex\u00e1gono.<br \/>\n\u00ad<\/li>\n<li>Comecemos por determinar a altura de um desses tri\u00e2ngulos:<br \/>\n\\[\\overline {OM} = \\sqrt {{{\\overline {OB} }^2} &#8211; {{\\overline {OM} }^2}} = \\sqrt {{5^2} &#8211; {{\\left( {{\\textstyle{5 \\over 2}}} \\right)}^2}} = \\sqrt {\\frac{{100 &#8211; 25}}{4}} = \\sqrt {\\frac{{3 \\times 25}}{4}} = \\frac{{5\\sqrt 3 }}{2}\\]<br \/>\nPortanto, a \u00e1rea\u00a0do hex\u00e1gono, em cent\u00edmetros quadrados com arredondamento \u00e0 unidade, \u00e9:<br \/>\n\\[{A_{\\left[ {ABCDEF} \\right]}} = 6 \\times {A_{\\left[ {OBC} \\right]}} = 6 \\times \\frac{{\\overline {BC} \\times \\overline {OM} }}{2} = 6 \\times \\frac{{5 \\times {\\textstyle{{5\\sqrt 3 } \\over 2}}}}{2} = \\frac{{75\\sqrt 3 }}{2} \\approx 65\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_13453' onClick='GTTabs_show(0,13453)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Desenha um hex\u00e1gono regular inscrito numa circunfer\u00eancia e os raios correspondentes aos extremos de cada um dos lados do hex\u00e1gono. Classifica cada um dos tri\u00e2ngulos obtidos. Que rela\u00e7\u00e3o existe entre o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20502,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,278],"tags":[426,279,188],"series":[],"class_list":["post-13453","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-circunferencia-e-poligonos","tag-9-o-ano","tag-angulo-ao-centro","tag-circunferencia"],"views":5868,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/02\/9V1Pag145-16_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13453","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=13453"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13453\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20502"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13453"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13453"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13453"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=13453"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}