{"id":3701,"date":"2010-10-07T22:46:12","date_gmt":"2010-10-07T21:46:12","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=3701"},"modified":"2022-01-21T01:36:53","modified_gmt":"2022-01-21T01:36:53","slug":"rascunho-3","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=3701","title":{"rendered":"Um pent\u00e1gono"},"content":{"rendered":"<p><ul id='GTTabs_ul_3701' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_3701' class='GTTabs_curr'><a  id=\"3701_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_3701' ><a  id=\"3701_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_3701'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"3713\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=3713\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono.jpg\" data-orig-size=\"347,337\" 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\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono.jpg\" class=\"alignright size-medium wp-image-3713\" title=\"Pent\u00e1gono\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono-300x291.jpg\" alt=\"\" width=\"168\" height=\"163\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono-300x291.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono-150x145.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono.jpg 347w\" sizes=\"auto, (max-width: 168px) 100vw, 168px\" \/><\/a>Na figura est\u00e1 representado um c\u00edrculo trigonom\u00e9trico de centro O e um pent\u00e1gono regular [ABCDO].<\/p>\n<ol>\n<li>Mostre que as coordenadas de B s\u00e3o $(1-\\cos \\alpha ;sen\\,72{}^\\text{o})$.<\/li>\n<li>Determine uma express\u00e3o para obter a \u00e1rea do pent\u00e1gono em fun\u00e7\u00e3o de $\\alpha $ (em graus) e apresente um valor dessa \u00e1rea aproximada \u00e0s d\u00e9cimas.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_3701' onClick='GTTabs_show(1,3701)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_3701'>\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\/pentagono.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"3713\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=3713\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono.jpg\" data-orig-size=\"347,337\" 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\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono.jpg\" class=\"alignright size-medium wp-image-3713\" title=\"Pent\u00e1gono\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono-300x291.jpg\" alt=\"\" width=\"210\" height=\"204\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono-300x291.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono-150x145.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pentagono.jpg 347w\" sizes=\"auto, (max-width: 210px) 100vw, 210px\" \/><\/a>O pent\u00e1gono regular pode ser obtido pela divis\u00e3o da circunfer\u00eancia que o circunscreve em cinco arcos geometricamente iguais.<br \/>\nCada um dos seus \u00e2ngulos internos \u00e9 um \u00e2ngulo inscrito nessa circunfer\u00eancia, logo a sua amplitude \u00e9 \\(\\alpha = \\frac{{A\\mathord{\\buildrel{\\lower3pt\\hbox{$\\scriptscriptstyle\\frown$}} \\over B} D}}{2} = \\frac{{3 \\times \\frac{{{{360}^{\\rm{o}}}}}{5}}}{2} = {108^{\\rm{o}}}\\).<br \/>\nO ponto D tem coordenadas $({{x}_{D}},{{y}_{D}})$, sendo ${{x}_{D}}=\\cos \\alpha $ e ${{y}_{D}}=sen\\,\\alpha $ (note: ${{x}_{D}}&lt;0\\wedge {{y}_{D}}&gt;0$).<br \/>\n\u00c9 \u00fatil, neste momento, reparar que o pent\u00e1gono \u00e9 sim\u00e9trico em rela\u00e7\u00e3o \u00e0 recta que cont\u00e9m C e \u00e9 paralela ao eixo das ordenadas.<br \/>\nAssim, os pontos D e B t\u00eam iguais ordenadas, logo ${{y}_{B}}={{y}_{D}}=sen\\,\\alpha $.<br \/>\nPor outro lado, sendo D&#8217;e B&#8217; as proje\u00e7\u00f5es ortogonais dos pontos D e B, respetivamente, sobre o eixo das abcissas, tem-se que \\[\\overline{D&#8217;O}=\\overline{AB&#8217;}=-{{x}_{D}}=-\\cos \\alpha \\] Logo, ${{x}_{B}}=\\overline{OA}+\\overline{AB&#8217;}=1-\\cos \\alpha $.<br \/>\nVimos acima que ${{y}_{B}}={{y}_{D}}=sen\\,\\alpha $ e que $\\alpha =108{}^\\text{o}$, logo\u00a0\\[{{y}_{B}}={{y}_{D}}=sen\\,108{}^\\text{o}=sen\\,(180{}^\\text{o}-108{}^\\text{o})=sen\\,72{}^\\text{o}\\]Conclui-se, como quer\u00edamos mostrar, que \\[B\\,(1-\\cos \\alpha ;sen\\,72{}^\\text{o})\\]<br \/>\n\u00ad<\/li>\n<li>Seja C&#8217; a proje\u00e7\u00e3o ortogonal de C sobre a recta BD.<br \/>\nOra, o tri\u00e2ngulo [DCC&#8217;] \u00e9 ret\u00e2ngulo em C&#8217; e $D\\widehat{C}C&#8217;=\\frac{D\\widehat{C}B}{2}=\\frac{108{}^\\text{o}}{2}=54{}^\\text{o}$.<br \/>\nLogo, $\\cos 54{}^\\text{o}=\\frac{\\overline{CC&#8217;}}{\\overline{DC}}=\\frac{\\overline{CC&#8217;}}{1}=\\overline{CC&#8217;}$.<br \/>\nDecompondo o pent\u00e1gono num trap\u00e9zio e num tri\u00e2ngulo, ambos is\u00f3sceles, temos:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\nA &amp; = &amp; {{A}_{[ABDO]}}+{{A}_{[BCD]}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\overline{DB}+\\overline{OA}}{2}\\times \\overline{BA&#8217;}+\\frac{\\overline{DB}\\times \\overline{CC&#8217;}}{2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{(-\\cos \\alpha +1-\\cos \\alpha )+1}{2}\\times sen\\,\\alpha +\\frac{(-\\cos \\alpha +1-\\cos \\alpha )\\times \\cos 54{}^\\text{o}}{2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; (1-\\cos \\alpha )\\times sen\\,\\alpha +\\frac{(1-2\\cos \\alpha )}{2}\\times \\cos 54{}^\\text{o}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nComo $\\alpha =108{}^\\text{o}$, vem: \\[\\begin{array}{*{35}{l}}<br \/>\nA(\\alpha =108{}^\\text{o}) &amp; = &amp; (1-\\cos 108{}^\\text{o})\\times sen\\,108{}^\\text{o}+\\frac{(1-2\\cos 108{}^\\text{o})}{2}\\times \\cos 54{}^\\text{o}\u00a0 \\\\<br \/>\n{} &amp; \\simeq\u00a0 &amp; 1,72\u00a0 \\\\<br \/>\n\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_3701' onClick='GTTabs_show(0,3701)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e1 representado um c\u00edrculo trigonom\u00e9trico de centro O e um pent\u00e1gono regular [ABCDO]. Mostre que as coordenadas de B s\u00e3o $(1-\\cos \\alpha ;sen\\,72{}^\\text{o})$. Determine uma express\u00e3o para obter a&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20796,"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-3701","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":2820,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/11V1Pag093-35_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/3701","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=3701"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/3701\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20796"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3701"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3701"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3701"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=3701"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}