{"id":6397,"date":"2010-12-19T21:36:46","date_gmt":"2010-12-19T21:36:46","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6397"},"modified":"2022-01-21T23:28:47","modified_gmt":"2022-01-21T23:28:47","slug":"uma-circunferencia","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6397","title":{"rendered":"Uma circunfer\u00eancia"},"content":{"rendered":"<p><ul id='GTTabs_ul_6397' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6397' class='GTTabs_curr'><a  id=\"6397_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6397' ><a  id=\"6397_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_6397'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6398\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6398\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60.jpg\" data-orig-size=\"324,334\" 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=\"Circunfer\u00eancia\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60.jpg\" class=\"alignright wp-image-6398\" title=\"Circunfer\u00eancia\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60-291x300.jpg\" alt=\"\" width=\"240\" height=\"247\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60-291x300.jpg 291w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60-145x150.jpg 145w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60.jpg 324w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Na figura ao lado est\u00e3o representados:<\/p>\n<ul>\n<li>uma circunfer\u00eancia de centro O e raio 1 unidade de comprimento;<\/li>\n<li>um di\u00e2metro [AB] e uma corda [CD], perpendicular a esse di\u00e2metro.<\/li>\n<\/ul>\n<p>Designando por $\\alpha $ a amplitude do \u00e2ngulo AOC, em radianos:<\/p>\n<ol>\n<li>determine o valor de $\\overrightarrow{AB}\\,.\\,\\overrightarrow{CD}$;<\/li>\n<li>mostre que a \u00e1rea do tri\u00e2ngulo [BCD] em fun\u00e7\u00e3o de $\\alpha $ \u00e9 $A(\\alpha )=sen\\,\\alpha \\times (1+\\cos \\alpha )$;<\/li>\n<li>determine o valor da \u00e1rea do tri\u00e2ngulo quando\u00a0$\\overline{OM}$ \u00e9 igual a $\\overline{MC}$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6397' onClick='GTTabs_show(1,6397)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6397'>\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\/12\/pag-188-60.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6398\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6398\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60.jpg\" data-orig-size=\"324,334\" 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=\"Circunfer\u00eancia\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60.jpg\" class=\"alignright wp-image-6398\" title=\"Circunfer\u00eancia\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60-291x300.jpg\" alt=\"\" width=\"240\" height=\"247\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60-291x300.jpg 291w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60-145x150.jpg 145w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60.jpg 324w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Como $AB\\bot CD$, os vetores s\u00e3o perpendiculares e, por isso, $\\overrightarrow{AB}\\,.\\,\\overrightarrow{CD}=0$.<br \/>\n\u00ad<\/li>\n<li>Como $\\cos \\alpha =\\frac{\\overline{MO}}{\\overline{OC}}$ e $sen\\,\\alpha =\\frac{\\overline{MC}}{\\overline{OC}}$, temos: $\\overline{MO}=\\cos \\alpha $ e $\\overline{MC}=sen\\,\\alpha $, pois $\\overline{OC}=1$.\n<p>Assim, temos: \\[\\begin{array}{*{35}{l}}<br \/>\nA(\\alpha ) &amp; = &amp; \\frac{\\overline{CD}\\times \\overline{MB}}{2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{2\\times sen\\,\\alpha \\times (\\cos \\alpha +1)}{2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; sen\\,\\alpha \\times (\\cos \\alpha +1)\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Quando $\\overline{OM}$ \u00e9 igual a $\\overline{MC}$, o tri\u00e2ngulo ret\u00e2ngulo [OMC] \u00e9 is\u00f3sceles. Logo, $\\alpha =\\frac{\\pi }{4}$ radianos.\n<p>Assim, vem:\u00a0\\[A(\\frac{\\pi }{4})=sen\\,\\frac{\\pi }{4}\\times (\\cos \\frac{\\pi }{4}+1)=\\frac{\\sqrt{2}}{2}(\\frac{\\sqrt{2}}{2}+1)=\\frac{1+\\sqrt{2}}{2}\\]\u00a0<a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6398\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6398\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60.jpg\" data-orig-size=\"324,334\" 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=\"Circunfer\u00eancia\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60.jpg\" class=\"aligncenter wp-image-6398\" title=\"Circunfer\u00eancia\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60-291x300.jpg\" alt=\"\" width=\"240\" height=\"247\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60-291x300.jpg 291w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60-145x150.jpg 145w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-60.jpg 324w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a><\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6397' onClick='GTTabs_show(0,6397)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura ao lado est\u00e3o representados: uma circunfer\u00eancia de centro O e raio 1 unidade de comprimento; um di\u00e2metro [AB] e uma corda [CD], perpendicular a esse di\u00e2metro. Designando por $\\alpha&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20845,"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],"series":[],"class_list":["post-6397","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"],"views":2650,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag188-60_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6397","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=6397"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6397\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20845"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6397"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6397"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6397"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6397"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}