{"id":23595,"date":"2022-12-09T23:54:34","date_gmt":"2022-12-09T23:54:34","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=23595"},"modified":"2022-12-27T23:58:25","modified_gmt":"2022-12-27T23:58:25","slug":"o-triangulo-abc-e-retangulo-em-a","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=23595","title":{"rendered":"O tri\u00e2ngulo [ABC] \u00e9 ret\u00e2ngulo em A"},"content":{"rendered":"\n<p><ul id='GTTabs_ul_23595' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_23595' class='GTTabs_curr'><a  id=\"23595_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_23595' ><a  id=\"23595_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_23595'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>O tri\u00e2ngulo [ABC] \u00e9 ret\u00e2ngulo em A, sendo [AP] a altura referente \u00e0 hipotenusa.<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23598\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23598\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1.png\" data-orig-size=\"339,199\" 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_Pag068-1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1.png\" class=\"aligncenter wp-image-23598 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1-300x176.png\" alt=\"\" width=\"300\" height=\"176\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1-300x176.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1.png 339w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p>Sabendo que \\(\\overline {AP} = 8\\) cm e \\(\\overline {CP} = 10\\) cm, qual \u00e9 :<\/p>\n<ol>\n<li>o comprimento de [BP]?<br \/><strong>[A]<\/strong> 6,4 cm\u00a0 \u00a0 \u00a0 <strong>[B]<\/strong> 10 cm\u00a0 \u00a0 \u00a0 <strong>[C]<\/strong> 7 cm\u00a0 \u00a0 \u00a0 <strong>[D]<\/strong> 8,2 cm<\/li>\n<li>o comprimento, aproximado \u00e0s d\u00e9cimas, de cada um dos catetos de [ABC]?<br \/><strong>[A]<\/strong> 6,4 cm e 19,3 cm.<br \/><strong>[B]<\/strong> 10,2 cm e 12,8 cm.<br \/><strong>[C]<\/strong> 10,2 cm e 19,3 cm.<br \/><strong>[D]<\/strong> 12,8 cm e 19,3 cm.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_23595' onClick='GTTabs_show(1,23595)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_23595'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>O tri\u00e2ngulo [ABC] \u00e9 ret\u00e2ngulo em A, sendo [AP] a altura referente \u00e0 hipotenusa.<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23600\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23600\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos.png\" data-orig-size=\"339,199\" 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_Pag068-1_Angulos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos.png\" class=\"aligncenter wp-image-23600 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos-300x176.png\" alt=\"\" width=\"300\" height=\"176\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos-300x176.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos.png 339w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p><span style=\"font-size: revert; color: initial;\">Sabendo que \\(\\overline {AP} = 8\\) cm e \\(\\overline {CP} = 10\\) cm, qual \u00e9 :<\/span><\/p>\n<\/blockquote>\n<ol>\n<li>\n<blockquote>o comprimento de [BP]?<br \/><strong>[A]<\/strong> 6,4 cm\u00a0 \u00a0 \u00a0 <strong>[B]<\/strong> 10 cm\u00a0 \u00a0 \u00a0 <strong>[C]<\/strong> 7 cm\u00a0 \u00a0 \u00a0 <strong>[D]<\/strong> 8,2 cm<\/blockquote>\n<\/li>\n<li>\n<blockquote>o comprimento, aproximado \u00e0s d\u00e9cimas, de cada um dos catetos de [ABC]?<br \/><strong>[A]<\/strong> 6,4 cm e 19,3 cm.<br \/><strong>[B]<\/strong> 10,2 cm e 12,8 cm.<br \/><strong>[C]<\/strong> 10,2 cm e 19,3 cm.<br \/><strong>[D]<\/strong> 12,8 cm e 19,3 cm.<\/blockquote>\n<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23600\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23600\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos.png\" data-orig-size=\"339,199\" 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_Pag068-1_Angulos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos.png\" class=\"size-medium wp-image-23600 alignright\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos-300x176.png\" alt=\"\" width=\"300\" height=\"176\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos-300x176.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos.png 339w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/>Como os tri\u00e2ngulos [ACP] e [ABP] s\u00e3o semelhantes (crit\u00e9rio AA), ent\u00e3o os comprimentos dos lados correspondentes s\u00e3o diretamente proporcionais:<br \/>\\[\\frac{{\\overline {AP} }}{{\\overline {BP} }} = \\frac{{\\overline {CP} }}{{\\overline {AP} }} = \\frac{{\\overline {AC} }}{{\\overline {AB} }}\\]<br \/>Considerando as duas primeiras raz\u00f5es e substituindo os valores conhecidos, vem:<br \/>\\[\\begin{array}{*{20}{l}}{\\frac{8}{{\\overline {BP} }} = \\frac{{10}}{8}}&amp; \\Leftrightarrow &amp;{10 \\times \\overline {BP} = 8 \\times 8}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {BP} = \\frac{{8 \\times 8}}{{10}}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {BP} = 6,4}\\end{array}\\]<br \/>Portanto, a op\u00e7\u00e3o correta \u00e9 <strong>[A]<\/strong> 6,4 cm.<br \/><br \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"23600\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=23600\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos.png\" data-orig-size=\"339,199\" 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_Pag068-1_Angulos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos.png\" class=\"size-medium wp-image-23600 alignright\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos-300x176.png\" alt=\"\" width=\"300\" height=\"176\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos-300x176.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_Angulos.png 339w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/>Aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [ACP], vem:<br \/>\\[\\overline {AC} = \\sqrt {{{\\overline {AP} }^2} + {{\\overline {CP} }^2}} = \\sqrt {{8^2} + {{10}^2}} = \\sqrt {164} \\approx 12,8\\]<br \/>Aplicando agora o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [ABP], vem:<br \/>\\[\\overline {AB} = \\sqrt {{{\\overline {AP} }^2} + {{\\overline {BP} }^2}} = \\sqrt {{8^2} + {{6,4}^2}} = \\sqrt {104,96} \\approx 10,2\\]<br \/>Portanto, a op\u00e7\u00e3o correta \u00e9\u00a0<strong>[B]<\/strong> 10,2 cm e 12,8 cm.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_23595' onClick='GTTabs_show(0,23595)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado O tri\u00e2ngulo [ABC] \u00e9 ret\u00e2ngulo em A, sendo [AP] a altura referente \u00e0 hipotenusa. Sabendo que \\(\\overline {AP} = 8\\) cm e \\(\\overline {CP} = 10\\) cm, qual \u00e9 : o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":23599,"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-23595","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":235,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2022\/12\/8_Pag068-1_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/23595","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=23595"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/23595\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/23599"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=23595"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=23595"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=23595"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=23595"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}