{"id":23074,"date":"2022-11-17T19:27:08","date_gmt":"2022-11-17T19:27:08","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=23074"},"modified":"2022-11-17T20:15:53","modified_gmt":"2022-11-17T20:15:53","slug":"a-distancia-entre-as-arvores","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=23074","title":{"rendered":"A dist\u00e2ncia entre as \u00e1rvores"},"content":{"rendered":"\n<p><ul id='GTTabs_ul_23074' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_23074' class='GTTabs_curr'><a  id=\"23074_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_23074' ><a  id=\"23074_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_23074'>\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_T2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23076\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23076\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2.png\" data-orig-size=\"960,504\" 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_T2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2.png\" class=\"alignright wp-image-23076\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2-300x158.png\" alt=\"\" width=\"440\" height=\"231\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2-300x158.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2-768x403.png 768w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2.png 960w\" sizes=\"auto, (max-width: 440px) 100vw, 440px\" \/><\/a>Observa na figura o procedimento usado pela Marta para descobrir a dist\u00e2ncia entre as \u00e1rvores que se encontram nos pontos A e B.<\/p>\n<p>A medida do comprimento do seu passo \u00e9 80 cm.<\/p>\n<ol>\n<li>Justifica que os tri\u00e2ngulos [ABC] e [ADE] s\u00e3o semelhantes.<\/li>\n<li>Qual \u00e9, em metros, a dist\u00e2ncia entre as \u00e1rvores que se encontram nos pontos A e B?<br \/>Explica a tua resposta.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_23074' onClick='GTTabs_show(1,23074)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_23074'>\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_T2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23076\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23076\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2.png\" data-orig-size=\"960,504\" 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_T2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2.png\" class=\"alignright wp-image-23076\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2-300x158.png\" alt=\"\" width=\"440\" height=\"231\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2-300x158.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2-768x403.png 768w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2.png 960w\" sizes=\"auto, (max-width: 440px) 100vw, 440px\" \/><\/a>Observa na figura o procedimento usado pela Marta para descobrir a dist\u00e2ncia entre as \u00e1rvores que se encontram nos pontos A e B.<\/p>\n<p>A medida do comprimento do seu passo \u00e9 80 cm.<\/p>\n<\/blockquote>\n<ol>\n<li>\n<blockquote>Justifica que os tri\u00e2ngulos [ABC] e [ADE] s\u00e3o semelhantes.<\/blockquote>\n<\/li>\n<li>\n<blockquote>Qual \u00e9, em metros, a dist\u00e2ncia entre as \u00e1rvores que se encontram nos pontos A e B?<br \/>Explica a tua resposta.<\/blockquote>\n<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>Pelo crit\u00e9rio AA, os tri\u00e2ngulos [ABC] e [ADE] s\u00e3o semelhantes, pois \\(C\\widehat AB = D\\widehat AE\\) (\u00e2ngulos verticalmente opostos) e \\(A\\widehat CB = A\\widehat DE\\) (\u00e2ngulos retos).<br \/><br \/><\/li>\n<li>Como os tri\u00e2ngulos s\u00e3o semelhantes, ent\u00e3o os comprimentos dos lados correspondentes s\u00e3o diretamente proporcionais:<br \/>\\[\\frac{{\\overline {AB} }}{{\\overline {AE} }} = \\frac{{\\overline {AC} }}{{\\overline {AD} }} = \\frac{{\\overline {BC} }}{{\\overline {DE} }}\\]<br \/>Selecionando as duas raz\u00f5es mais \u00e0 esquerda, \\(\\frac{{\\overline {AB} }}{{\\overline {AE} }} = \\frac{{\\overline {AC} }}{{\\overline {AD} }}\\), e substituindo os valores conhecidos, vem:<br \/>\\[\\begin{array}{*{20}{l}}{\\frac{{\\overline {AB} }}{{30}} = \\frac{{60}}{{25}}}&amp; \\Leftrightarrow &amp;{25 \\times \\overline {AB} = 30 \\times 60}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {AB} = \\frac{{1800}}{{\\mathop {25}\\limits_{\\left( 4 \\right)} }}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {AB} = \\frac{{7200}}{{100}}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {AB} = 72}\\end{array}\\]<br \/>Ora, \\(72\\;{\\rm{passos}} = 72 \\times 0,8\\;{\\rm{m}} = 57,6\\;{\\rm{m}}\\).<br \/>Logo, a dist\u00e2ncia entre as \u00e1rvores que se encontram nos pontos A e B \u00e9 57,6 metros.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_23074' onClick='GTTabs_show(0,23074)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Observa na figura o procedimento usado pela Marta para descobrir a dist\u00e2ncia entre as \u00e1rvores que se encontram nos pontos A e B. A medida do comprimento do seu passo \u00e9&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":23077,"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-23074","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":234,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/11\/8_Pag047_T2-520x245-1.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/23074","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=23074"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/23074\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/23077"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=23074"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=23074"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=23074"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=23074"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}