{"id":4677,"date":"2010-10-25T16:43:34","date_gmt":"2010-10-25T15:43:34","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=4677"},"modified":"2022-01-21T02:39:45","modified_gmt":"2022-01-21T02:39:45","slug":"um-poligono-abeg","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=4677","title":{"rendered":"Um pol\u00edgono [ABEG]"},"content":{"rendered":"<p><ul id='GTTabs_ul_4677' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_4677' class='GTTabs_curr'><a  id=\"4677_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_4677' ><a  id=\"4677_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_4677'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag99-72.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"4682\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=4682\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag99-72.jpg\" data-orig-size=\"371,229\" 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=\"Pol\u00edgono\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag99-72.jpg\" class=\"alignright wp-image-4682\" title=\"Pol\u00edgono\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag99-72-300x185.jpg\" alt=\"\" width=\"320\" height=\"198\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag99-72-300x185.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag99-72-150x92.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag99-72.jpg 371w\" sizes=\"auto, (max-width: 320px) 100vw, 320px\" \/><\/a>Na figura est\u00e1 representado, a cor, um pol\u00edgono [ABEG].<br \/>\nTem-se que:<\/p>\n<ul>\n<li>[ABFG] \u00e9 um quadrado de lado 2.<\/li>\n<li>FD \u00e9 um arco de circunfer\u00eancia de centro em B; o ponto E move-se ao longo desse arco; em consequ\u00eancia, o ponto C desloca-se sobre o segmento [BD], de tal forma que se tem sempre $[EC]\\bot [BD]$.<\/li>\n<li>x designa a amplitude, em radianos, do \u00e2ngulo CBE $\\left( x\\in \\left[ 0,\\frac{\\pi }{2} \\right] \\right)$.<\/li>\n<\/ul>\n<ol>\n<li>Mostre que a \u00e1rea do pol\u00edgono [ABEG] \u00e9 dada, em fun\u00e7\u00e3o de x, por $A(x)=2(1+sen\\,x+\\cos x)$. (Sugest\u00e3o: pode ser-lhe \u00fatil considerar o trap\u00e9zio [ACEG])<\/li>\n<li>Determine $A(0)$ e $A(\\frac{\\pi }{2})$. Interprete, geometricamente, cada um dos valores obtidos.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_4677' onClick='GTTabs_show(1,4677)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_4677'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><span class=\"alignright\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"text-align: right\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":414,\r\n\"height\":276,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49  50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 0,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/span>No tri\u00e2ngulo ret\u00e2ngulo [BCE], obt\u00e9m-se: \\[\\cos x=\\frac{\\overline{BC}}{2}\\Leftrightarrow \\overline{BC}=2\\cos x\\] e \\[sen\\,x=\\frac{\\overline{EC}}{2}\\Leftrightarrow \\overline{EC}=2\\,sen\\,x\\] Logo, \\[\\begin{array}{*{35}{l}} \u00a0\u00a0 A(x) &amp; = &amp; {{A}_{[ACEG]}}-{{A}_{[BCE]}}\u00a0 \\\\ \u00a0\u00a0 {} &amp; = &amp; \\frac{\\overline{AG}+\\overline{CE}}{2}\\times \\overline{AC}-\\frac{\\overline{BC}\\times \\overline{CE}}{2}\u00a0 \\\\ \u00a0\u00a0 {} &amp; = &amp; \\frac{2+2\\,sen\\,x}{2}\\times (2+2\\cos x)-\\frac{2\\cos x\\times 2\\,sen\\,x}{2}\u00a0 \\\\ \u00a0\u00a0 {} &amp; = &amp; (1+sen\\,x)\\times (2+2\\cos x)-2\\cos x\\times \\,sen\\,x\u00a0 \\\\ \u00a0\u00a0 {} &amp; = &amp; 2+2\\cos x+2\\,sen\\,x+2\\cos x\\times sen\\,x-2\\cos x\\times \\,sen\\,x\u00a0 \\\\ \u00a0\u00a0 {} &amp; = &amp; 2(1+\\cos x+sen\\,x)\u00a0 \\\\ \\end{array}\\]<br \/>\n\u00ad<\/li>\n<li>Os valores pedidos s\u00e3o: \\[A(0)=2(1+\\cos 0+sen\\,0)=2(1+1+0)=4\\] \\[A(\\frac{\\pi }{2})=2(1+\\cos \\frac{\\pi }{2}+sen\\,\\frac{\\pi }{2})=2(1+0+1)=4\\]<br \/>\nPara $x=0$, o ponto E coincide com o ponto D.<br \/>\nNesta situa\u00e7\u00e3o, o pol\u00edgono colorido\u00a0\u00e9 o tri\u00e2ngulo [ADG], cuja \u00e1rea \u00e9 igual a 4.<\/p>\n<p>Para $x=\\frac{\\pi }{2}$, o ponto E coincide com o ponto F.<br \/>\nNesta situa\u00e7\u00e3o, o pol\u00edgono colorido \u00e9 o quadrado [ABFG], cuja \u00e1rea \u00e9, tamb\u00e9m, \u00a0igual a 4.<br \/>\n(Explore a anima\u00e7\u00e3o acima)<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_4677' onClick='GTTabs_show(0,4677)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e1 representado, a cor, um pol\u00edgono [ABEG]. Tem-se que: [ABFG] \u00e9 um quadrado de lado 2. FD \u00e9 um arco de circunfer\u00eancia de centro em B; o ponto E&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20808,"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-4677","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":3099,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/11V1Pag099-72_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/4677","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=4677"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/4677\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20808"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4677"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4677"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4677"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=4677"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}