{"id":12994,"date":"2017-11-24T21:51:03","date_gmt":"2017-11-24T21:51:03","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=12994"},"modified":"2022-01-17T18:40:00","modified_gmt":"2022-01-17T18:40:00","slug":"apotema-de-um-hexagono-regular","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=12994","title":{"rendered":"Ap\u00f3tema de um hex\u00e1gono regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_12994' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_12994' class='GTTabs_curr'><a  id=\"12994_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_12994' ><a  id=\"12994_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<li id='GTTabs_li_2_12994' ><a  id=\"12994_2\" onMouseOver=\"GTTabsShowLinks('A Prova'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>A Prova<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_12994'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2017\/11\/hexagono.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"12999\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12999\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2017\/11\/hexagono.png\" data-orig-size=\"418,383\" 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\/2017\/11\/hexagono.png\" class=\"alignright wp-image-12999 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2017\/11\/hexagono-300x275.png\" alt=\"\" width=\"300\" height=\"275\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2017\/11\/hexagono-300x275.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2017\/11\/hexagono.png 418w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Um hex\u00e1gono regular est\u00e1 inscrito numa circunfer\u00eancia de raio 6 cm.<\/p>\n<p>Determina o valor exato da medida do comprimento de um ap\u00f3tema do hex\u00e1gono.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_12994' onClick='GTTabs_show(1,12994)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_12994'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2017\/11\/hexagono.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"12999\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12999\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2017\/11\/hexagono.png\" data-orig-size=\"418,383\" 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\/2017\/11\/hexagono.png\" class=\"alignright wp-image-12999 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2017\/11\/hexagono-300x275.png\" alt=\"\" width=\"300\" height=\"275\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2017\/11\/hexagono-300x275.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2017\/11\/hexagono.png 418w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Seja G a proje\u00e7\u00e3o ortogonal do ponto O sobre o segmento de reta [EF].<\/p>\n<p>Como o hex\u00e1gono \u00e9 regular, ent\u00e3o os seis tri\u00e2ngulos em que est\u00e1 dividido s\u00e3o equil\u00e1teros e geometricamente iguais entre si. [Prove que assim \u00e9!]<\/p>\n<p>Assim, aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [EGO], temos:<\/p>\n<p>\\[\\overline {GO} = \\sqrt {{{\\overline {OE} }^2} &#8211; {{\\overline {EG} }^2}} = \\sqrt {{6^2} &#8211; {3^2}} = \\sqrt {36 &#8211; 9} = \\sqrt {27} = 3\\sqrt 3 \\;cm\\]<\/p>\n<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_12994' onClick='GTTabs_show(0,12994)'>&lt;&lt; Enunciado<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_12994' onClick='GTTabs_show(2,12994)'>A Prova &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_12994'>\n<span class='GTTabs_titles'><b>A Prova<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/HexagonoRegular2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7394\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7394\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/HexagonoRegular2.png\" data-orig-size=\"262,274\" 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;}\" data-image-title=\"Hexagono regular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/HexagonoRegular2.png\" class=\"alignright size-full wp-image-7394\" title=\"Hexagono regular\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/HexagonoRegular2.png\" alt=\"\" width=\"262\" height=\"274\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/HexagonoRegular2.png 262w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/HexagonoRegular2-143x150.png 143w\" sizes=\"auto, (max-width: 262px) 100vw, 262px\" \/><\/a>Comecemos por provar que o tri\u00e2ngulo [AOB] \u00e9 equil\u00e1tero.<\/p>\n<p>[OA] e [OB] s\u00e3o raios da mesma circunfer\u00eancia, logo o tri\u00e2ngulo [AOB] \u00e9 is\u00f3sceles.<\/p>\n<p>Num tri\u00e2ngulo, a lados geometricamente iguais op\u00f5em-se \u00e2ngulos geometricamente iguais. Logo, os \u00e2ngulos OAB e OBA s\u00e3o geometricamente iguais.<\/p>\n<p>Como a amplitude do \u00e2ngulo ao centro AOB \u00e9 60\u00ba, pois \u00e9 igual \u00e0 amplitude de cada um dos 6 arcos geometricamente iguais em que a circunfer\u00eancia foi dividida, resulta que os tr\u00eas \u00e2ngulos internos do tri\u00e2ngulo [AOB] s\u00e3o geometricamente iguais.<\/p>\n<p>Sendo equi\u00e2ngulo, o tri\u00e2ngulo [AOB] \u00e9 equil\u00e1tero.<\/p>\n<p>E, como a circunfer\u00eancia foi dividida em seis arcos geometricamente iguais, conclui-se que os seis tri\u00e2ngulos em que o hex\u00e1gono est\u00e1 dividido s\u00e3o geometricamente iguais entre si e equil\u00e1teros.<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_12994' onClick='GTTabs_show(1,12994)'>&lt;&lt; Resolu\u00e7\u00e3o<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um hex\u00e1gono regular est\u00e1 inscrito numa circunfer\u00eancia de raio 6 cm. Determina o valor exato da medida do comprimento de um ap\u00f3tema do hex\u00e1gono. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":20519,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,187],"tags":[426,451,452],"series":[],"class_list":["post-12994","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-8--ano","category-aplicando","category-lugares-geometricos","tag-9-o-ano","tag-apotema","tag-hexagono"],"views":3056,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2017\/11\/9V1Pag087-T2_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/12994","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=12994"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/12994\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20519"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=12994"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=12994"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=12994"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=12994"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}