{"id":7369,"date":"2012-02-11T23:04:34","date_gmt":"2012-02-11T23:04:34","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7369"},"modified":"2022-01-16T20:04:37","modified_gmt":"2022-01-16T20:04:37","slug":"quatro-angulos-internos-de-um-quadrilatero","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7369","title":{"rendered":"Quatro \u00e2ngulos internos de um quadril\u00e1tero"},"content":{"rendered":"<p><ul id='GTTabs_ul_7369' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7369' class='GTTabs_curr'><a  id=\"7369_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7369' ><a  id=\"7369_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_7369'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Sabendo que PA e PB s\u00e3o tangentes \u00e0 circunfer\u00eancia e que $\\mathop {AB}\\limits^\\frown\u00a0\u00a0 = 140^\\circ $, determina a amplitude dos quatro \u00e2ngulos internos do quadril\u00e1tero [OAPB].<\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7370\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7370\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2.jpg\" data-orig-size=\"490,286\" 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=\"Quadril\u00e1tero\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2.jpg\" class=\"aligncenter wp-image-7370\" title=\"Quadril\u00e1tero\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2.jpg\" alt=\"\" width=\"350\" height=\"204\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2.jpg 490w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2-300x175.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2-150x87.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2-400x233.jpg 400w\" sizes=\"auto, (max-width: 350px) 100vw, 350px\" \/><\/a><\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7369' onClick='GTTabs_show(1,7369)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7369'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7370\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7370\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2.jpg\" data-orig-size=\"490,286\" 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=\"Quadril\u00e1tero\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2.jpg\" class=\"alignright wp-image-7370\" title=\"Quadril\u00e1tero\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2.jpg\" alt=\"\" width=\"350\" height=\"204\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2.jpg 490w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2-300x175.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2-150x87.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/pag23-2-400x233.jpg 400w\" sizes=\"auto, (max-width: 350px) 100vw, 350px\" \/><\/a>A reta tangente a uma circunfer\u00eancia \u00e9 perpendicular \u00e0 reta que cont\u00e9m o centro da circunfer\u00eancia e o ponto de tang\u00eancia.<\/p>\n<p>Logo, os \u00e2ngulos OAP e OBP s\u00e3o retos, pelo que $O\\widehat AP = O\\widehat BP = 90^\\circ $.<\/p>\n<p>O \u00e2ngulo AOB \u00e9 um \u00e2ngulo ao centro, pelo que a sua amplitude \u00e9 igual \u00e0 do arco compreendido entre os seus lados.<\/p>\n<p>Portanto, $A\\widehat OB = \\mathop {AB}\\limits^\\frown\u00a0\u00a0 = 140^\\circ $.<\/p>\n<p>Como a soma das amplitudes dos \u00e2ngulos internos de um quadril\u00e1tero convexo \u00e9 360\u00ba, temos:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{A\\widehat PB}&amp; = &amp;{360^\\circ\u00a0 &#8211; (P\\widehat AO + A\\widehat OB + O\\widehat BP)} \\\\<br \/>\n{}&amp; = &amp;{360^\\circ\u00a0 &#8211; (90^\\circ\u00a0 + 140^\\circ\u00a0 + 90^\\circ )} \\\\<br \/>\n{}&amp; = &amp;{40^\\circ }<br \/>\n\\end{array}$$<\/p>\n<p>\u00ad<\/p>\n<blockquote>\n<h3 style=\"text-align: center;\">\u00a0Soma das amplitudes dos \u00e2ngulos internos de um quadril\u00e1tero<\/h3>\n<\/p>\n<p style=\"text-align: center;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":809,\r\n\"height\":495,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 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 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 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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': 1,'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><\/p>\n<\/p>\n<p>Comecemos por dividir o quadril\u00e1tero [ABCD] em dois tri\u00e2ngulos por interm\u00e9dio da diagonal [BD].<\/p>\n<p>Tendo em considera\u00e7\u00e3o que a soma das amplitudes dos \u00e2ngulos internos de um tri\u00e2ngulo \u00e9 180\u00ba, temos:<\/p>\n<p>$$\\begin{array}{*{20}{c}}<br \/>\n{\\alpha\u00a0 + {\\beta _1} + {\\delta _1}}&amp; = &amp;{180^\\circ } \\\\<br \/>\n{{\\beta _2} + \\gamma\u00a0 + {\\delta _2}}&amp; = &amp;{180^\\circ } \\\\<br \/>\n\\hline<br \/>\n{\\alpha\u00a0 + ({\\beta _1} + {\\beta _2}) + \\gamma\u00a0 + ({\\delta _1} + {\\delta _2})}&amp; = &amp;{180^\\circ\u00a0 + 180^\\circ } \\\\<br \/>\n{\\alpha\u00a0 + \\beta\u00a0 + \\gamma\u00a0 + \\delta }&amp; = &amp;{360^\\circ }<br \/>\n\\end{array}$$<\/p>\n<\/blockquote>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7369' onClick='GTTabs_show(0,7369)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sabendo que PA e PB s\u00e3o tangentes \u00e0 circunfer\u00eancia e que $\\mathop {AB}\\limits^\\frown\u00a0\u00a0 = 140^\\circ $, determina a amplitude dos quatro \u00e2ngulos internos do quadril\u00e1tero [OAPB]. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":20413,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,278],"tags":[426,279,280,188,282],"series":[],"class_list":["post-7369","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-ao-centro","tag-angulo-inscrito","tag-circunferencia","tag-tangente-a-uma-circunferencia"],"views":3257,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/02\/9V1Pag032-2_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7369","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=7369"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7369\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20413"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7369"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7369"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7369"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7369"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}