{"id":23461,"date":"2022-12-08T10:06:01","date_gmt":"2022-12-08T10:06:01","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=23461"},"modified":"2022-12-18T22:22:58","modified_gmt":"2022-12-18T22:22:58","slug":"dois-triangulos-retangulos","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=23461","title":{"rendered":"Dois tri\u00e2ngulos ret\u00e2ngulos"},"content":{"rendered":"\n<p><ul id='GTTabs_ul_23461' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_23461' class='GTTabs_curr'><a  id=\"23461_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_23461' ><a  id=\"23461_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_23461'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag064-3.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23463\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23463\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag064-3.png\" data-orig-size=\"235,319\" 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_Pag064-3\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag064-3.png\" class=\"alignright wp-image-23463 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag064-3-221x300.png\" alt=\"\" width=\"221\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag064-3-221x300.png 221w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag064-3.png 235w\" sizes=\"auto, (max-width: 221px) 100vw, 221px\" \/><\/a>Na figura, est\u00e3o representados dois tri\u00e2ngulos, [ABC] e [EDC], ret\u00e2ngulos, respetivamente em A e D, sendo E e D pontos, respetivamente, dos segmentos de reta [AC] e [BC].<\/p>\n<ol>\n<li>Justifica que os tri\u00e2ngulos s\u00e3o semelhantes.<\/li>\n<li>Supondo que \\(\\overline {CB} = 10\\) cm, \\(\\overline {CE} = 5\\) cm e que \\(\\overline {DE} = 3\\) cm, determina:<\/li>\n<\/ol>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>a raz\u00e3o de semelhan\u00e7a que aplica o tri\u00e2ngulo [CDE] no tri\u00e2ngulo [CAB].<\/li>\n<li>a medida de \\(\\overline {CD} \\).<\/li>\n<li>as medidas de \\(\\overline {AC} \\) e \\(\\overline {AB} \\).<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_23461' onClick='GTTabs_show(1,23461)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_23461'>\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\/12\/8_Pag064-3.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23463\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23463\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag064-3.png\" data-orig-size=\"235,319\" 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_Pag064-3\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag064-3.png\" class=\"alignright wp-image-23463 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag064-3-221x300.png\" alt=\"\" width=\"221\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag064-3-221x300.png 221w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag064-3.png 235w\" sizes=\"auto, (max-width: 221px) 100vw, 221px\" \/><\/a>Na figura, est\u00e3o representados dois tri\u00e2ngulos, [ABC] e [EDC], ret\u00e2ngulos, respetivamente em A e D, sendo E e D pontos, respetivamente, dos segmentos de reta [AC] e [BC].<\/p>\n<\/blockquote>\n<ol>\n<li>\n<blockquote>Justifica que os tri\u00e2ngulos s\u00e3o semelhantes.<\/blockquote>\n<\/li>\n<li>\n<blockquote>Supondo que \\(\\overline {CB} = 10\\) cm, \\(\\overline {CE} = 5\\) cm e que \\(\\overline {DE} = 3\\) cm, determina:<\/blockquote>\n<\/li>\n<\/ol>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>\n<blockquote>a raz\u00e3o de semelhan\u00e7a que aplica o tri\u00e2ngulo [CDE] no tri\u00e2ngulo [CAB].<\/blockquote>\n<\/li>\n<li>\n<blockquote>a medida de \\(\\overline {CD} \\).<\/blockquote>\n<\/li>\n<li>\n<blockquote>as medidas de \\(\\overline {AC} \\) e \\(\\overline {AB} \\).<\/blockquote>\n<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>Os tri\u00e2ngulos [ABC] e [EDC] s\u00e3o semelhantes, pois ambos possuem um \u00e2ngulo reto e o \u00e2ngulo ACB \u00e9 comum aos dois tri\u00e2ngulos &#8211; crit\u00e9rio AA.<br \/><br \/><\/li>\n<li>\u00a0<\/li>\n<\/ol>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Como os tri\u00e2ngulos [ABC] e [EDC] s\u00e3o semelhantes, ent\u00e3o os comprimentos dos lados correspondentes s\u00e3o diretamente proporcionais:<br \/>\\[\\frac{{\\overline {AB} }}{{\\overline {DE} }} = \\frac{{\\overline {AC} }}{{\\overline {CD} }} = \\frac{{\\overline {BC} }}{{\\overline {CE} }}\\]<br \/>Substituindo os valores conhecidos na express\u00e3o anterior, vem:<br \/>\\[\\frac{{\\overline {AB} }}{3} = \\frac{{\\overline {AC} }}{{\\overline {CD} }} = \\frac{{10}}{5}\\]<br \/>Portanto, a raz\u00e3o de semelhan\u00e7a que aplica o tri\u00e2ngulo [CDE] no tri\u00e2ngulo [CAB] \u00e9 \\(r = \\frac{{\\overline {BC} }}{{\\overline {CE} }} = \\frac{{10}}{5} = 2\\).<br \/><br \/><\/li>\n<li>Aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [CDE], temos:<br \/>\\[\\overline {CD} = \\sqrt {{{\\overline {CE} }^2} &#8211; {{\\overline {DE} }^2}} = \\sqrt {{5^2} &#8211; {3^2}} = \\sqrt {25 &#8211; 9} = \\sqrt {16} = 4\\]<\/li>\n<li>Como a raz\u00e3o de semelhan\u00e7a\u00a0que aplica o tri\u00e2ngulo [CDE] no tri\u00e2ngulo [CAB] \u00e9 \\(r = 2\\), temos:<br \/>\\[\\begin{array}{*{20}{c}}{\\frac{{\\overline {AB} }}{{\\overline {DE} }} = 2}&amp; \\Leftrightarrow &amp;{\\frac{{\\overline {AB} }}{3} = 2}&amp; \\Leftrightarrow &amp;{\\overline {AB} = 6}\\end{array}\\]<br \/>\\[\\begin{array}{*{20}{c}}{\\frac{{\\overline {AC} }}{{\\overline {CD} }} = 2}&amp; \\Leftrightarrow &amp;{\\frac{{\\overline {AC} }}{4} = 2}&amp; \\Leftrightarrow &amp;{\\overline {AC} = 8}\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_23461' onClick='GTTabs_show(0,23461)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura, est\u00e3o representados dois tri\u00e2ngulos, [ABC] e [EDC], ret\u00e2ngulos, respetivamente em A e D, sendo E e D pontos, respetivamente, dos segmentos de reta [AC] e [BC]. Justifica que os&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":23464,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,682],"tags":[424,67,149,118],"series":[],"class_list":["post-23461","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-geometria","tag-semelhanca-de-triangulos","tag-teorema-de-pitagoras"],"views":233,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag064-3_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/23461","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=23461"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/23461\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/23464"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=23461"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=23461"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=23461"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=23461"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}