{"id":13336,"date":"2018-01-25T16:59:38","date_gmt":"2018-01-25T16:59:38","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=13336"},"modified":"2022-01-17T01:00:46","modified_gmt":"2022-01-17T01:00:46","slug":"uma-circunferencia-e-duas-semirretas-tangentes","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=13336","title":{"rendered":"Uma circunfer\u00eancia e duas semirretas tangentes"},"content":{"rendered":"<p><ul id='GTTabs_ul_13336' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_13336' class='GTTabs_curr'><a  id=\"13336_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_13336' ><a  id=\"13336_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_13336'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/01\/9V1Pag135-7.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13337\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13337\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/01\/9V1Pag135-7.png\" data-orig-size=\"350,260\" 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\/01\/9V1Pag135-7.png\" class=\"alignright size-medium wp-image-13337\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/01\/9V1Pag135-7-300x223.png\" alt=\"\" width=\"300\" height=\"223\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/01\/9V1Pag135-7-300x223.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/01\/9V1Pag135-7.png 350w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Na figura, est\u00e1 representada uma circunfer\u00eancia de centro O e duas semirretas, concorrentes em V, e que s\u00e3o tangentes \u00e0 circunfer\u00eancia nos pontos A e B.<\/p>\n<ol>\n<li>Justifica que \\(\\overline {VA} = \\overline {VB} \\).<\/li>\n<li>Supondo que a amplitude do arco AB mede 120 graus, determina a medida da amplitude do \u00e2ngulo AVB.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_13336' onClick='GTTabs_show(1,13336)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_13336'>\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\/2018\/01\/9V1Pag135-7.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13337\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13337\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/01\/9V1Pag135-7.png\" data-orig-size=\"350,260\" 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\/01\/9V1Pag135-7.png\" class=\"alignright size-medium wp-image-13337\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/01\/9V1Pag135-7-300x223.png\" alt=\"\" width=\"300\" height=\"223\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/01\/9V1Pag135-7-300x223.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/01\/9V1Pag135-7.png 350w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>A reta tangente a uma circunfer\u00eancia \u00e9 perpendicular ao raio dirigido ao ponto de tang\u00eancia, pelo que os \u00e2ngulos OAV e OBV s\u00e3o retos. Portanto, os tri\u00e2ngulos [AOV] e [BOV] s\u00e3o tri\u00e2ngulos ret\u00e2ngulos, com iguais hipotenusas, o segmento de reta [OV], bem como um cateto igual, os segmentos [AO] e [BO], que s\u00e3o raios da mesma circunfer\u00eancia.<br \/>\nAssim, por aplica\u00e7\u00e3o do Teorema de Pit\u00e1goras em ambos os tri\u00e2ngulos ret\u00e2ngulos, resulta \\(\\overline {VA} = \\overline {VB} \\).<br \/>\n\u00ad<\/li>\n<li>Tendo em considera\u00e7\u00e3o que o \u00e2ngulo AVB \u00e9 um \u00e2ngulo de v\u00e9rtice exterior a um c\u00edrculo, vem (ADB \u00e9 o arco maior AB):<br \/>\n\\[A\\widehat VB = \\frac{{\\overparen{ADB} &#8211; \\overparen{AB}}}{2} = \\frac{{240^\\circ &#8211; 120^\\circ }}{2} = 60^\\circ \\]<br \/>\nALTERNATIVA: Determine as amplitudes dos \u00e2ngulos internos do tri\u00e2ngulo [AOV] e conclua sobre a amplitude do \u00e2ngulo AVB.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_13336' onClick='GTTabs_show(0,13336)'>&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 e duas semirretas, concorrentes em V, e que s\u00e3o tangentes \u00e0 circunfer\u00eancia nos pontos A e B. Justifica que \\(\\overline {VA} =&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20469,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,278],"tags":[426,462,280,188],"series":[],"class_list":["post-13336","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-circunferencia-e-poligonos","tag-9-o-ano","tag-angulo-de-vertice-exterior-a-um-circulo","tag-angulo-inscrito","tag-circunferencia"],"views":1932,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/01\/9V1Pag135-7_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13336","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=13336"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13336\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20469"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13336"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13336"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13336"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=13336"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}