{"id":11867,"date":"2014-05-01T23:01:41","date_gmt":"2014-05-01T22:01:41","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11867"},"modified":"2022-01-22T16:42:09","modified_gmt":"2022-01-22T16:42:09","slug":"a-dobra-numa-folha-de-papel","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11867","title":{"rendered":"A dobra numa folha de papel"},"content":{"rendered":"<p><ul id='GTTabs_ul_11867' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11867' class='GTTabs_curr'><a  id=\"11867_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11867' ><a  id=\"11867_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_11867'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/11-2-pag177-10.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11868\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11868\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/11-2-pag177-10.png\" data-orig-size=\"671,524\" 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;}\" data-image-title=\"Folha de papel\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/11-2-pag177-10.png\" class=\"aligncenter wp-image-11868\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/11-2-pag177-10.png\" alt=\"Folha de papel\" width=\"250\" height=\"195\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/11-2-pag177-10.png 671w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/11-2-pag177-10-300x234.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/11-2-pag177-10-150x117.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/11-2-pag177-10-400x312.png 400w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/><\/a>Considere uma folha de papel retangular de comprimento 24 unidades e largura 18 unidades.<br \/>\nDobramos a folha de papel de modo que o v\u00e9rtice A coincida com o v\u00e9rtice C e vincamos a folha.<\/p>\n<p>Qual \u00e9 o comprimento do vinco?<\/p>\n<ul style=\"list-style-type: square;\">\n<li><strong>Sugest\u00e3o<\/strong>: Comece por dobrar uma folha de papel retangular e descubra as rela\u00e7\u00f5es entre os v\u00e1rios elementos geom\u00e9tricos.<\/li>\n<\/ul>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11867' onClick='GTTabs_show(1,11867)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11867'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><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\":426,\r\n\"height\":436,\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 , 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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>De acordo com os elementos registados na imagem\u00a0acima, depois de dobrada a folha de papel, constata-se que que o tri\u00e2ngulo [BPC] \u00e9 ret\u00e2ngulo em B, sendo $\\overline {PC}\u00a0 = 24 &#8211; x$.<\/p>\n<p>\u00a0Como $\\overline {PC}\u00a0 = \\sqrt {{{\\overline {PB} }^2} + {{\\overline {BC} }^2}} $, vem:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{24 &#8211; x = \\sqrt {{x^2} + {{18}^2}} }&amp; \\Rightarrow &amp;{{{\\left( {24 &#8211; x} \\right)}^2} = {x^2} + {{18}^2}} \\\\<br \/>\n{}&amp; \\Rightarrow &amp;{576 &#8211; 48x + {x^2} = {x^2} + 324} \\\\<br \/>\n{}&amp; \\Rightarrow &amp;{x = 5,25}<br \/>\n\\end{array}\\]<\/p>\n<\/p>\n<ul style=\"list-style-type: square;\">\n<li>Verifica\u00e7\u00e3o da solu\u00e7\u00e3o: $24 &#8211; 5,25 = \\sqrt {{{5,25}^2} + {{18}^2}}\u00a0 \\Leftrightarrow 18,75 = 18,75$.<\/li>\n<\/ul>\n<p>Tendo em considera\u00e7\u00e3o que [PQ] \u00e9 a hipotenusa de um tri\u00e2ngulo ret\u00e2ngulo de catetos com $18$ e $24 &#8211; 2 \\times 5,25$ unidades de comprimento, vem:<\/p>\n<p>\\[v = \\overline {PQ}\u00a0 = \\sqrt {{{18}^2} + {{\\left( {24 &#8211; 2 \\times 5,25} \\right)}^2}}\u00a0 = \\sqrt {324 + 182,5}\u00a0 = 22,5\\]<\/p>\n<p>\u00a0Portanto, o vinco tem $22,5$ unidades de comprimento.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11867' onClick='GTTabs_show(0,11867)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere uma folha de papel retangular de comprimento 24 unidades e largura 18 unidades. Dobramos a folha de papel de modo que o v\u00e9rtice A coincida com o v\u00e9rtice C e&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20898,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,157],"tags":[422,158],"series":[],"class_list":["post-11867","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-com-radicais","tag-11-o-ano","tag-funcoes-com-radicais-2"],"views":1868,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/11V2Pag177-10_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11867","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=11867"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11867\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20898"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11867"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11867"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11867"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11867"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}