{"id":5254,"date":"2010-11-12T02:33:15","date_gmt":"2010-11-12T02:33:15","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5254"},"modified":"2022-01-21T16:46:13","modified_gmt":"2022-01-21T16:46:13","slug":"abcde-e-um-pentagono-regular","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5254","title":{"rendered":"[ABCDE] \u00e9 um pent\u00e1gono regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_5254' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5254' class='GTTabs_curr'><a  id=\"5254_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5254' ><a  id=\"5254_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_5254'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"5262\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=5262\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12.jpg\" data-orig-size=\"319,329\" 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=\"Pent\u00e1gono regular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12.jpg\" class=\"alignright wp-image-5262\" title=\"Pent\u00e1gono regular\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12-290x300.jpg\" alt=\"\" width=\"240\" height=\"248\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12-290x300.jpg 290w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12-145x150.jpg 145w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12.jpg 319w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>[ABCDE] \u00e9 um pent\u00e1gono regular de lado <em>l<\/em>, inscrito na circunfer\u00eancia de centro O e raio <em>r<\/em>.<\/p>\n<ol>\n<li>Calcule em fun\u00e7\u00e3o de <em>r<\/em>, com aproxima\u00e7\u00e3o \u00e0s cent\u00e9simas:\n<p>$\\overrightarrow{OA}.\\overrightarrow{OB}$,\u00a0$\\overrightarrow{OA}.\\overrightarrow{OE}$ e\u00a0$\\overrightarrow{OA}.\\overrightarrow{OC}$.<\/p>\n<\/li>\n<li>Determine, em fun\u00e7\u00e3o de <em>l<\/em>, com aproxima\u00e7\u00e3o \u00e0s cent\u00e9simas:\n<p>$\\overrightarrow{AB}.\\overrightarrow{AE}$, $\\overrightarrow{AB}.\\overrightarrow{ED}$, $\\overrightarrow{AB}.\\overrightarrow{AD}$ e $\\overrightarrow{AB}.\\overrightarrow{AC}$.<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5254' onClick='GTTabs_show(1,5254)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5254'>\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\/11\/pag178-12.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"5262\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=5262\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12.jpg\" data-orig-size=\"319,329\" 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=\"Pent\u00e1gono regular\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12.jpg\" class=\"alignright wp-image-5262\" title=\"Pent\u00e1gono regular\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12-290x300.jpg\" alt=\"\" width=\"240\" height=\"248\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12-290x300.jpg 290w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12-145x150.jpg 145w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag178-12.jpg 319w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>A circunfer\u00eancia foi dividida em cinco arcos geometricamente iguais, de amplitude $\\frac{360{}^\\text{o}}{5}=72{}^\\text{o}$.\n<p>$\\overrightarrow{OA}.\\overrightarrow{OB}=\\left\\| \\overrightarrow{OA} \\right\\|.\\left\\| \\overrightarrow{OB} \\right\\|.\\cos (\\widehat{\\overrightarrow{OA}\\,\\overrightarrow{OB}})=r\\times r\\times \\cos 72{}^\\text{o}\\simeq 0,31\\times {{r}^{2}}$<\/p>\n<p>$\\overrightarrow{OA}.\\overrightarrow{OE}=\\left\\| \\overrightarrow{OA} \\right\\|.\\left\\| \\overrightarrow{OE} \\right\\|.\\cos (\\widehat{\\overrightarrow{OA}\\,\\overrightarrow{OE}})=r\\times r\\times \\cos 72{}^\\text{o}\\simeq 0,31\\times {{r}^{2}}$<\/p>\n<p>$\\overrightarrow{OA}.\\overrightarrow{OC}=\\left\\| \\overrightarrow{OA} \\right\\|.\\left\\| \\overrightarrow{OC} \\right\\|.\\cos (\\widehat{\\overrightarrow{OA}\\,\\overrightarrow{OC}})=r\\times r\\times \\cos 144{}^\\text{o}\\simeq -0,81\\times {{r}^{2}}$<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>A amplitude do \u00e2ngulo interno do pent\u00e1gono regular \u00e9 $\\frac{3\\times 72{}^\\text{o}}{2}=108{}^\\text{o}$.<\/p>\n<p>$\\overrightarrow{AB}.\\overrightarrow{AE}=\\left\\| \\overrightarrow{AB} \\right\\|.\\left\\| \\overrightarrow{AE} \\right\\|.\\cos (\\widehat{\\overrightarrow{AB}\\,\\overrightarrow{AE}})=l\\times l\\times \\cos 108{}^\\text{o}\\simeq -0,31\\times {{l}^{2}}$<\/p>\n<p>$\\overrightarrow{AB}.\\overrightarrow{ED}=\\left\\| \\overrightarrow{AB} \\right\\|.\\left\\| \\overrightarrow{ED} \\right\\|.\\cos (\\widehat{\\overrightarrow{AB}\\,\\overrightarrow{ED}})=l\\times l\\times \\cos 36{}^\\text{o}\\simeq 0,81\\times {{l}^{2}}$<br \/>\n(As semi-rectas $\\dot{B}A$ e $\\dot{D}E$ intersectam-se no ponto P, sendo $A\\hat{P}E=36{}^\\text{o}$ (Porqu\u00ea?))<\/p>\n<p>$\\overrightarrow{AB}.\\overrightarrow{AD}=\\left\\| \\overrightarrow{AB} \\right\\|.\\left\\| \\overrightarrow{AD} \\right\\|.\\cos (\\widehat{\\overrightarrow{AB}\\,\\overrightarrow{AD}})=l\\times l\\times \\cos 72{}^\\text{o}\\simeq 0,31\\times {{l}^{2}}$<\/p>\n<p>$\\overrightarrow{AB}.\\overrightarrow{AC}=\\left\\| \\overrightarrow{AB} \\right\\|.\\left\\| \\overrightarrow{AC} \\right\\|.\\cos (\\widehat{\\overrightarrow{AB}\\,\\overrightarrow{AC}})=l\\times l\\times \\cos 36{}^\\text{o}\\simeq 0,81\\times {{l}^{2}}$<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5254' onClick='GTTabs_show(0,5254)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado [ABCDE] \u00e9 um pent\u00e1gono regular de lado l, inscrito na circunfer\u00eancia de centro O e raio r. Calcule em fun\u00e7\u00e3o de r, com aproxima\u00e7\u00e3o \u00e0s cent\u00e9simas: $\\overrightarrow{OA}.\\overrightarrow{OB}$,\u00a0$\\overrightarrow{OA}.\\overrightarrow{OE}$ e\u00a0$\\overrightarrow{OA}.\\overrightarrow{OC}$. Determine, em fun\u00e7\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20825,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67,111],"series":[],"class_list":["post-5254","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-geometria-analitica","tag-11-o-ano","tag-geometria","tag-vectores"],"views":3091,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/11V1Pag178-12_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5254","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=5254"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5254\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20825"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5254"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5254"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5254"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5254"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}