{"id":23064,"date":"2022-11-17T18:29:02","date_gmt":"2022-11-17T18:29:02","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=23064"},"modified":"2022-11-17T19:22:45","modified_gmt":"2022-11-17T19:22:45","slug":"a-distancia-que-separa-a-joana-da-arvore","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=23064","title":{"rendered":"A dist\u00e2ncia que separa a Joana da \u00e1rvore"},"content":{"rendered":"\n<p><ul id='GTTabs_ul_23064' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_23064' class='GTTabs_curr'><a  id=\"23064_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_23064' ><a  id=\"23064_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_23064'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23066\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23066\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1.png\" data-orig-size=\"800,540\" 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=\"8_pag047-T1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1.png\" class=\"alignright wp-image-23066\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1-300x203.png\" alt=\"\" width=\"400\" height=\"270\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1-300x203.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1-768x518.png 768w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1.png 800w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a>A corda fixada nos pontos A e B e a corda fixada nos pontos C e D s\u00e3o paralelas.<\/p>\n<p>Determina a dist\u00e2ncia que separa a Joana (ponto A) da \u00e1rvore que est\u00e1 na outra margem do rio.<br \/>Explica a tua resposta.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_23064' onClick='GTTabs_show(1,23064)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_23064'>\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\/2022\/11\/8_pag047-T1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23066\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23066\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1.png\" data-orig-size=\"800,540\" 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=\"8_pag047-T1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1.png\" class=\"alignright wp-image-23066\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1-300x203.png\" alt=\"\" width=\"400\" height=\"270\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1-300x203.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1-768x518.png 768w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1.png 800w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a>A corda fixada nos pontos A e B e a corda fixada nos pontos C e D s\u00e3o paralelas.<\/p>\n<p>Determina a dist\u00e2ncia que separa a Joana (ponto A) da \u00e1rvore que est\u00e1 na outra margem do rio.<br \/>Explica a tua resposta.<\/p>\n<\/blockquote>\n<p><br \/>Designemos o <em>p\u00e9<\/em> da \u00e1rvore, que est\u00e1 na outra margem, pelo ponto P.<\/p>\n<p>Ora, de acordo com o Teorema de Tales, temos:<\/p>\n<p>\\[\\frac{{\\overline {PA} }}{{\\overline {PC} }} = \\frac{{\\overline {AB} }}{{\\overline {CD} }} = \\frac{{\\overline {PB} }}{{\\overline {PD} }}\\]<\/p>\n<p>Selecionando as duas raz\u00f5es mais \u00e0 esquerda, \\(\\frac{{\\overline {PA} }}{{\\overline {PC} }} = \\frac{{\\overline {AB} }}{{\\overline {CD} }}\\), e substituindo os valores conhecidos, vem:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{\\frac{{\\overline {PA} }}{{\\overline {PA} + 2}} = \\frac{{40}}{{45}}}&amp; \\Leftrightarrow &amp;{45 \\times \\overline {PA} = 40 \\times \\left( {\\overline {PA} + 2} \\right)}\\\\{}&amp; \\Leftrightarrow &amp;{45 \\times \\overline {PA} = 40 \\times \\overline {PA} + 80}\\\\{}&amp; \\Leftrightarrow &amp;{45 \\times \\overline {PA} &#8211; 40 \\times \\overline {PA} = 80}\\\\{}&amp; \\Leftrightarrow &amp;{5 \\times \\overline {PA} = 80}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {PA} = \\frac{{80}}{5}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {PA} = 16}\\end{array}\\]<\/p>\n<p>Portanto, a dist\u00e2ncia da Joana \u00e0 \u00e1rvore \u00e9 16 metros.<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_23064' onClick='GTTabs_show(0,23064)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A corda fixada nos pontos A e B e a corda fixada nos pontos C e D s\u00e3o paralelas. Determina a dist\u00e2ncia que separa a Joana (ponto A) da \u00e1rvore que&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":23067,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,682],"tags":[424,149,683],"series":[],"class_list":["post-23064","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-8--ano","category-aplicando","category-teorema-de-pitagoras","tag-8-o-ano","tag-semelhanca-de-triangulos","tag-teorema-de-tales"],"views":162,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_pag047-T1_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/23064","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=23064"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/23064\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/23067"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=23064"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=23064"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=23064"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=23064"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}