{"id":9916,"date":"2012-10-03T18:55:39","date_gmt":"2012-10-03T17:55:39","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9916"},"modified":"2022-01-20T21:45:17","modified_gmt":"2022-01-20T21:45:17","slug":"unindo-os-pontos-medios-dos-lados-de-um-quadrilatero","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9916","title":{"rendered":"Unindo os pontos m\u00e9dios dos lados de um quadril\u00e1tero \u2013 Parte 1"},"content":{"rendered":"<p><ul id='GTTabs_ul_9916' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9916' class='GTTabs_curr'><a  id=\"9916_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9916' ><a  id=\"9916_1\" onMouseOver=\"GTTabsShowLinks('R1 e R2'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>R1 e R2<\/a><\/li>\n<li id='GTTabs_li_2_9916' ><a  id=\"9916_2\" onMouseOver=\"GTTabsShowLinks('R3'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>R3<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_9916'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Determine tr\u00eas, ou mais, quadril\u00e1teros como os que se seguem.<\/p>\n<\/p>\n<div id=\"attachment_9920\" style=\"width: 665px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-9920\" data-attachment-id=\"9920\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9920\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros.jpg\" data-orig-size=\"655,178\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349287512&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\u00e1teros\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;Quadril\u00e1teros&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros.jpg\" class=\"size-full wp-image-9920\" title=\"Quadril\u00e1teros\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros.jpg\" alt=\"\" width=\"655\" height=\"178\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros.jpg 655w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros-300x81.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros-150x40.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros-400x108.jpg 400w\" sizes=\"auto, (max-width: 655px) 100vw, 655px\" \/><\/a><p id=\"caption-attachment-9920\" class=\"wp-caption-text\">Quadril\u00e1teros<\/p><\/div>\n<ol>\n<li>Determine os pontos m\u00e9dios dos lados dos quadril\u00e1teros e, em cada um deles, construa os segmentos de reta definidos por pontos m\u00e9dios de lados consecutivos.<\/li>\n<li>Investigue que tipo de quadril\u00e1teros obteve. Registe as suas conjeturas e tente justific\u00e1-las.<\/li>\n<li>Recorrendo a propriedades estudadas, prove, por exemplo, que o pol\u00edgono que se obt\u00e9m unindo os pontos m\u00e9dios dos lados consecutivos do quadril\u00e1tero [ABCD] \u00e9 um paralelogramo.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9916' onClick='GTTabs_show(1,9916)'>R1 e R2 &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9916'>\n<span class='GTTabs_titles'><b>R1 e R2<\/b><\/span><!--more--><\/p>\n<p>Determine tr\u00eas, ou mais, quadril\u00e1teros como os que se seguem.<\/p>\n<\/p>\n<div id=\"attachment_9920\" style=\"width: 665px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-9920\" data-attachment-id=\"9920\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9920\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros.jpg\" data-orig-size=\"655,178\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349287512&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\u00e1teros\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;Quadril\u00e1teros&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros.jpg\" class=\"size-full wp-image-9920\" title=\"Quadril\u00e1teros\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros.jpg\" alt=\"\" width=\"655\" height=\"178\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros.jpg 655w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros-300x81.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros-150x40.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadrilateros-400x108.jpg 400w\" sizes=\"auto, (max-width: 655px) 100vw, 655px\" \/><\/a><p id=\"caption-attachment-9920\" class=\"wp-caption-text\">Quadril\u00e1teros<\/p><\/div>\n<ol>\n<li>Determine os pontos m\u00e9dios dos lados dos quadril\u00e1teros e, em cada um deles, construa os segmentos de reta definidos por pontos m\u00e9dios de lados consecutivos.\n<p>FICA AO CUIDADO DO LEITOR &#8211; Utilize a aplica\u00e7\u00e3o apresentada abaixo.<\/p>\n<\/li>\n<li>Investigue que tipo de quadril\u00e1teros obteve. Registe as suas conjeturas e tente justific\u00e1-las.\n<p>FICA AO CUIDADO DO LEITOR &#8211; Utilize a aplica\u00e7\u00e3o apresentada abaixo.<\/p>\n<\/li>\n<\/ol>\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\":805,\r\n\"height\":482,\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': 0,'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><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9916' onClick='GTTabs_show(0,9916)'>&lt;&lt; Enunciado<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9916' onClick='GTTabs_show(2,9916)'>R3 &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_9916'>\n<span class='GTTabs_titles'><b>R3<\/b><\/span><\/p>\n<p>Vamos agora demonstrar que:<\/p>\n<blockquote>\n<p style=\"text-align: center;\"><strong>Os pontos m\u00e9dios dos lados consecutivos de qualquer quadril\u00e1tero s\u00e3o v\u00e9rtices de um paralelogramo.<\/strong><\/p>\n<\/blockquote>\n<table border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig1.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9934\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9934\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig1.jpg\" data-orig-size=\"314,274\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349297614&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=\"Fig.1\" data-image-description=\"&lt;p&gt;Figura 1&lt;\/p&gt;\n\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig1.jpg\" class=\"aligncenter  wp-image-9934\" title=\"Fig.1\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig1.jpg\" alt=\"\" width=\"251\" height=\"219\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig1.jpg 314w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig1-300x261.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig1-150x130.jpg 150w\" sizes=\"auto, (max-width: 251px) 100vw, 251px\" \/><\/a><\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig2.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9935\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9935\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig2.jpg\" data-orig-size=\"328,272\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349297663&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=\"Fig. 2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig2.jpg\" class=\"aligncenter  wp-image-9935\" title=\"Fig. 2\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig2.jpg\" alt=\"\" width=\"262\" height=\"218\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig2.jpg 328w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig2-300x248.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/fig2-150x124.jpg 150w\" sizes=\"auto, (max-width: 262px) 100vw, 262px\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>Figura 1<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Figura 2<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Na Figura 1, a diagonal [AC] dividiu o quadril\u00e1tero [ABCD] em dois tri\u00e2ngulos: [ABC] e [ADC].<\/p>\n<p>Considerem-se os tri\u00e2ngulos [ABC] e [MBN], relativamente aos quais se verifica:<\/p>\n<ul>\n<li>O \u00e2ngulo ABC \u00e9 comum aos dois tri\u00e2ngulos;<\/li>\n<li>Como M e N s\u00e3o pontos m\u00e9dios dos lados [AB] e [BC], tem-se: $$\\frac{{\\overline {AB} }}{{\\overline {MB} }} = 2 = \\frac{{\\overline {CB} }}{{\\overline {NB} }}$$<\/li>\n<\/ul>\n<p>Conclui-se, ent\u00e3o, que os tri\u00e2ngulos [ABC] e [MBN] s\u00e3o semelhantes, pois possuem dois pares de lados de comprimentos diretamente proporcionais e os \u00e2ngulos formados por esses pares de lados s\u00e3o geometricamente iguais (LAL).<\/p>\n<p>Consequentemente, nesses dois tri\u00e2ngulos, os \u00e2ngulos internos correspondentes s\u00e3o geometricamente iguais. Em particular, os \u00e2ngulos BMN e BAC s\u00e3o geometricamente iguais, pelo que os lados [MN] e [AC] s\u00e3o paralelos, isto \u00e9: $$\\left[ {MN} \\right]\\parallel \\left[ {AC} \\right]$$<\/p>\n<p>De forma an\u00e1loga, considerando agora os tri\u00e2ngulos semelhantes [ADC] e [QDP], conclui-se que tamb\u00e9m os lados [QP] e [AC] s\u00e3o paralelos, isto \u00e9: $$\\left[ {QP} \\right]\\parallel \\left[ {AC} \\right]$$<\/p>\n<p>Assim, sendo $\\left[ {MN} \\right]\\parallel \\left[ {AC} \\right]$ e $\\left[ {QP} \\right]\\parallel \\left[ {AC} \\right]$, ent\u00e3o [MN] e [QP] s\u00e3o paralelos, isto \u00e9: $$\\left[ {MN} \\right]\\parallel \\left[ {QP} \\right]$$<\/p>\n<p>Considerando agora a Figura 2 e a diagonal [BD] do quadril\u00e1tero [ABCD], por um processo an\u00e1logo ao utilizado relativamente \u00e0 Figura 1, conclui-se que [MQ] e [NP] s\u00e3o paralelos, isto \u00e9: $$\\left[ {MQ} \\right]\\parallel \\left[ {NP} \\right]$$<\/p>\n<\/p>\n<p>Portanto, sendo $\\left[ {MN} \\right]\\parallel \\left[ {QP} \\right]$ e $\\left[ {MQ} \\right]\\parallel \\left[ {NP} \\right]$, conclui-se que <strong>o quadril\u00e1tero [MNPQ] \u00e9 um paralelogramo<\/strong>, pois s\u00e3o paralelos os seus lados opostos.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9916' onClick='GTTabs_show(1,9916)'>&lt;&lt; R1 e R2<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado R1 e R2 Enunciado Determine tr\u00eas, ou mais, quadril\u00e1teros como os que se seguem. Determine os pontos m\u00e9dios dos lados dos quadril\u00e1teros e, em cada um deles, construa os segmentos de reta definidos&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20773,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,322],"tags":[429,67,430,149],"series":[],"class_list":["post-9916","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-10-o-ano","category-aplicando","category-modulo-inicial","tag-10-o-ano","tag-geometria","tag-modulo-inicial","tag-semelhanca-de-triangulos"],"views":6966,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/10V1Pag024-T-b_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9916","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=9916"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9916\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20773"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9916"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9916"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9916"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9916"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}