{"id":14285,"date":"2018-04-08T22:47:07","date_gmt":"2018-04-08T21:47:07","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14285"},"modified":"2022-01-11T13:58:24","modified_gmt":"2022-01-11T13:58:24","slug":"um-triangulo-inscrito-numa-circunferencia","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14285","title":{"rendered":"Um tri\u00e2ngulo inscrito numa circunfer\u00eancia"},"content":{"rendered":"<p><ul id='GTTabs_ul_14285' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14285' class='GTTabs_curr'><a  id=\"14285_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14285' ><a  id=\"14285_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_14285'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14287\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14287\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2.png\" data-orig-size=\"380,360\" 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=\"Circunfer\u00eancia\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2.png\" class=\"alignright size-medium wp-image-14287\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2-300x284.png\" alt=\"\" width=\"300\" height=\"284\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2-300x284.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2.png 380w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Na figura, est\u00e1 representada uma circunfer\u00eancia de centro no ponto <em>O<\/em>.<br \/>\nOs pontos <em>A<\/em>, <em>B<\/em>, <em>C<\/em>, <em>P<\/em> e <em>R<\/em> pertencem \u00e0 circunfer\u00eancia.<br \/>\nSabe-se que:<\/p>\n<ul>\n<li>a circunfer\u00eancia tem raio 8;<\/li>\n<li>\\(\\overline {BA} = \\overline {BC} \\);<\/li>\n<li>[<em>PR<\/em>] \u00e9 um di\u00e2metro da circunfer\u00eancia;<\/li>\n<li>o ponto <em>Q<\/em> \u00e9 o ponto de interse\u00e7\u00e3o dos segmentos [<em>BA<\/em>] e [<em>PR<\/em>];<\/li>\n<li>o ponto <em>S<\/em> \u00e9 o ponto de interse\u00e7\u00e3o dos segmentos [<em>BC<\/em>] e [<em>PR<\/em>];<\/li>\n<li>\\(A\\widehat BO = 36^\\circ \\);<\/li>\n<li>\\(B\\widehat OP = 90^\\circ \\).<\/li>\n<\/ul>\n<ol>\n<li>Qual \u00e9 a amplitude, em graus, do arco <em>AB<\/em>?<\/li>\n<li>Determina a \u00e1rea da regi\u00e3o representada a sombreado.<br \/>\nApresenta o resultado arredondado \u00e0s unidades.<br \/>\nApresenta os c\u00e1lculos que efetuares.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14285' onClick='GTTabs_show(1,14285)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14285'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14287\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14287\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2.png\" data-orig-size=\"380,360\" 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=\"Circunfer\u00eancia\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2.png\" class=\"alignright size-medium wp-image-14287\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2-300x284.png\" alt=\"\" width=\"300\" height=\"284\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2-300x284.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2.png 380w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Na figura, est\u00e1 representada uma circunfer\u00eancia de centro no ponto <em>O<\/em>.<br \/>\nOs pontos <em>A<\/em>, <em>B<\/em>, <em>C<\/em>, <em>P<\/em> e <em>R<\/em> pertencem \u00e0 circunfer\u00eancia.<br \/>\nSabe-se que:<\/p>\n<\/blockquote>\n<ul>\n<li>\n<blockquote>\n<p>a circunfer\u00eancia tem raio 8;<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>\\(\\overline {BA} = \\overline {BC} \\);<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>[<em>PR<\/em>] \u00e9 um di\u00e2metro da circunfer\u00eancia;<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>o ponto <em>Q<\/em> \u00e9 o ponto de interse\u00e7\u00e3o dos segmentos [<em>BA<\/em>] e [<em>PR<\/em>];<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>o ponto <em>S<\/em> \u00e9 o ponto de interse\u00e7\u00e3o dos segmentos [<em>BC<\/em>] e [<em>PR<\/em>];<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>\\(A\\widehat BO = 36^\\circ \\);<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>\\(B\\widehat OP = 90^\\circ \\).<\/p>\n<\/blockquote>\n<\/li>\n<\/ul>\n<ol>\n<li>Seja <em>B&#8217;<\/em> a imagem do ponto <em>B<\/em> na reflex\u00e3o de eixo <em>PR<\/em>.<br \/>\nOra, \\(\\overparen{AB} = \\overparen{BAB&#8217;} &#8211; \\overparen{AB&#8217;} = 180^\\circ &#8211; 2 \\times A\\widehat BB&#8217; = 180^\\circ &#8211; 2 \\times 36^\\circ = 108^\\circ \\).<br \/>\n\u00ad<\/li>\n<li>No tri\u00e2ngulo ret\u00e2ngulo [BOQ], vem:<br \/>\n\\[\\begin{array}{*{20}{l}}{{\\mathop{\\rm tg}\\nolimits} O\\widehat BQ = \\frac{{\\overline {OQ} }}{{\\overline {OB} }}}&amp; \\Leftrightarrow &amp;{{\\mathop{\\rm tg}\\nolimits} 36^\\circ = \\frac{{\\overline {OQ} }}{8}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {OQ} = 8 \\times {\\mathop{\\rm tg}\\nolimits} 36^\\circ }\\end{array}\\]<br \/>\nA \u00e1rea, em u.a.,\u00a0da regi\u00e3o representada a sombreado \u00e9:<br \/>\n\\[\\begin{array}{*{20}{l}}{{A_{Sombreada}}}&amp; = &amp;{\\frac{1}{2}{A_{C\u00edrculo}} &#8211; {A_{\\left[ {BQS} \\right]}}}\\\\{}&amp; = &amp;{\\frac{{\\pi \\times {8^2}}}{2} &#8211; \\frac{{\\overline {QS} \\times \\overline {OB} }}{2}}\\\\{}&amp; = &amp;{32 \\times \\pi &#8211; \\frac{{2 \\times 8 \\times {\\mathop{\\rm tg}\\nolimits} 36^\\circ \\times 8}}{2}}\\\\{}&amp; = &amp;{32 \\times \\pi &#8211; 64 \\times {\\mathop{\\rm tg}\\nolimits} 36^\\circ }\\\\{}&amp; \\approx &amp;{54}\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14285' onClick='GTTabs_show(0,14285)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura, est\u00e1 representada uma circunfer\u00eancia de centro no ponto O. Os pontos A, B, C, P e R pertencem \u00e0 circunfer\u00eancia. Sabe-se que: a circunfer\u00eancia tem raio 8; \\(\\overline {BA}&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14288,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,489],"tags":[426,280,491,493,490,492],"series":[],"class_list":["post-14285","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-trigonometria-9--ano","tag-9-o-ano","tag-angulo-inscrito","tag-cosseno","tag-razao-trigonometrica","tag-seno","tag-tangente"],"views":1812,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-2a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14285","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=14285"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14285\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14288"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14285"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14285"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14285"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14285"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}