{"id":14297,"date":"2018-04-09T16:22:45","date_gmt":"2018-04-09T15:22:45","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14297"},"modified":"2022-01-11T14:24:07","modified_gmt":"2022-01-11T14:24:07","slug":"outro-triangulo-inscrito-numa-circunferencia","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14297","title":{"rendered":"Outro tri\u00e2ngulo inscrito numa circunfer\u00eancia"},"content":{"rendered":"<p><ul id='GTTabs_ul_14297' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14297' class='GTTabs_curr'><a  id=\"14297_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14297' ><a  id=\"14297_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_14297'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-3.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14298\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14298\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-3.png\" data-orig-size=\"353,371\" 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\/9V2Pag66-3.png\" class=\"alignright wp-image-14298\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-3-285x300.png\" alt=\"\" width=\"240\" height=\"252\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-3-285x300.png 285w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-3.png 353w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Na figura, est\u00e1 representada uma circunfer\u00eancia, de centro <em>O<\/em>, em que:<\/p>\n<ul>\n<li><em>A<\/em>, <em>B<\/em>, <em>C<\/em> e <em>D<\/em> s\u00e3o pontos da circunfer\u00eancia;<\/li>\n<li>o segmento de reta [<em>BD<\/em>] \u00e9 um di\u00e2metro;<\/li>\n<li><em>E<\/em> \u00e9 o ponto de interse\u00e7\u00e3o das retas <em>BD<\/em> e <em>AC<\/em>;<\/li>\n<li>o tri\u00e2ngulo [<em>ADE<\/em>] \u00e9 ret\u00e2ngulo em <em>E<\/em>;<\/li>\n<li>\\(C\\widehat AD = 30^\\circ \\).<\/li>\n<\/ul>\n<ol>\n<li>Qual \u00e9 a amplitude, em graus, do arco <em>CD<\/em>?<\/li>\n<li>Sabendo que \\(\\overline {AD} = 5\\), determina \\(\\overline {ED} \\).<br \/>\nApresenta todos os c\u00e1lculos que efetuares.<\/li>\n<li>Sem efetuares medi\u00e7\u00f5es, explica por que \u00e9 que a seguinte afirma\u00e7\u00e3o \u00e9 verdadeira.<br \/>\n\u00abOs tri\u00e2ngulos [<em>ADE<\/em>] e [<em>CDE<\/em>] s\u00e3o geometricamente iguais].\u00bb<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14297' onClick='GTTabs_show(1,14297)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14297'>\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\/9V2Pag66-3.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14298\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14298\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-3.png\" data-orig-size=\"353,371\" 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\/9V2Pag66-3.png\" class=\"alignright wp-image-14298\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-3-285x300.png\" alt=\"\" width=\"240\" height=\"252\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-3-285x300.png 285w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-3.png 353w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Na figura, est\u00e1 representada uma circunfer\u00eancia, de centro <em>O<\/em>, em que:<\/p>\n<\/blockquote>\n<ul>\n<li>\n<blockquote>\n<p><em>A<\/em>, <em>B<\/em>, <em>C<\/em> e <em>D<\/em> s\u00e3o pontos da circunfer\u00eancia;<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>o segmento de reta [<em>BD<\/em>] \u00e9 um di\u00e2metro;<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p><em>E<\/em> \u00e9 o ponto de interse\u00e7\u00e3o das retas <em>BD<\/em> e <em>AC<\/em>;<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>o tri\u00e2ngulo [<em>ADE<\/em>] \u00e9 ret\u00e2ngulo em <em>E<\/em>;<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>\\(C\\widehat AD = 30^\\circ \\).<\/p>\n<\/blockquote>\n<\/li>\n<\/ul>\n<ol>\n<li>Tendo em considera\u00e7\u00e3o que <em>CAD<\/em> \u00e9 um \u00e2ngulo inscrito, vem: \\(\\overparen{CD} = 2 \\times C\\widehat AD = 2 \\times 30^\\circ = 60^\\circ \\).<br \/>\n\u00ad<\/li>\n<li>No tri\u00e2ngulo ret\u00e2ngulo [<em>ADE<\/em>], vem:<br \/>\n\\[\\begin{array}{*{20}{l}}{{\\mathop{\\rm sen}\\nolimits} C\\widehat AD = \\frac{{\\overline {ED} }}{{\\overline {AD} }}}&amp; \\Leftrightarrow &amp;{{\\mathop{\\rm sen}\\nolimits} 30^\\circ = \\frac{{\\overline {ED} }}{5}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {ED} = 5 \\times {\\mathop{\\rm sen}\\nolimits} 30^\\circ }\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {ED} = 2,5}\\end{array}\\]<\/li>\n<li>A reta <em>BD<\/em> \u00e9 a mediatriz do segmento [<em>AC<\/em>], pois \u00e9-lhe perpendicular e o ponto <em>O<\/em> \u00e9 equidistante dos extremos do segmento, visto [<em>OA<\/em>] e [<em>OC<\/em>] serem raios da mesma circunfer\u00eancia.<br \/>\nAssim, temos:<br \/>\n&#8211;\u00a0\\(\\left[ {AE} \\right] \\cong \\left[ {{\\rm{CE}}} \\right]\\);<br \/>\n&#8211;\u00a0\\(\\left[ {ED} \\right]\\) \u00e9 lado comum aos dois tri\u00e2ngulos;<br \/>\n&#8211;\u00a0\\(AED \\cong CED\\)<br \/>\nConsequentemente, pelo crit\u00e9rio LAL, os tri\u00e2ngulos [<em>ADE<\/em>] e [<em>CDE<\/em>] s\u00e3o geometricamente iguais.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14297' onClick='GTTabs_show(0,14297)'>&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 O, em que: A, B, C e D s\u00e3o pontos da circunfer\u00eancia; o segmento de reta [BD] \u00e9 um di\u00e2metro; E \u00e9 o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14300,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,489],"tags":[426,280,491,490,492],"series":[],"class_list":["post-14297","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-seno","tag-tangente"],"views":2951,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag66-3a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14297","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=14297"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14297\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14300"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14297"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14297"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14297"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14297"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}