{"id":14267,"date":"2018-04-03T18:17:28","date_gmt":"2018-04-03T17:17:28","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14267"},"modified":"2022-01-11T13:53:38","modified_gmt":"2022-01-11T13:53:38","slug":"um-triangulo-isosceles-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14267","title":{"rendered":"Um tri\u00e2ngulo is\u00f3sceles"},"content":{"rendered":"<p><ul id='GTTabs_ul_14267' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14267' class='GTTabs_curr'><a  id=\"14267_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14267' ><a  id=\"14267_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_14267'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag63-20.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14272\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14272\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag63-20.png\" data-orig-size=\"290,370\" 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=\"Tri\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag63-20.png\" class=\"alignright size-medium wp-image-14272\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag63-20-235x300.png\" alt=\"\" width=\"235\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag63-20-235x300.png 235w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag63-20.png 290w\" sizes=\"auto, (max-width: 235px) 100vw, 235px\" \/><\/a>Seja [<em>ABC<\/em>] um tri\u00e2ngulo tal que \\(\\overline {AC} = 4\\) e \\(\\overline {AB} = \\overline {BC} = 6\\).<\/p>\n<p>Seja <em>M<\/em> o ponto m\u00e9dio de [<em>AB<\/em>].<\/p>\n<p>Determina a medida da amplitude do \u00e2ngulo <em>ACM<\/em> com aproxima\u00e7\u00e3o \u00e0s d\u00e9cimas de grau, percorrendo as seguintes etapas.<\/p>\n<ol>\n<li>Tra\u00e7a a altura relativa ao v\u00e9rtice <em>B<\/em> e justifica que interseta [<em>AC<\/em>] no respetivo ponto m\u00e9dio <em>N<\/em>.<\/li>\n<li>Justifica que o ponto <em>Q<\/em>, interse\u00e7\u00e3o de [<em>MC<\/em>] com a altura relativa ao v\u00e9rtice <em>B<\/em>, \u00e9 o baricentro do tri\u00e2ngulo.<\/li>\n<li>Determina a medida de [<em>BN<\/em>] e de [<em>QN<\/em>].<\/li>\n<li>Utilizando uma raz\u00e3o trigonom\u00e9trica, determina a medida da amplitude do \u00e2ngulo <em>ACM<\/em>, com aproxima\u00e7\u00e3o \u00e0 d\u00e9cima de grau com o aux\u00edlio de uma calculadora.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14267' onClick='GTTabs_show(1,14267)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14267'>\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\/2018\/04\/9V2Pag63-20.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14272\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14272\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag63-20.png\" data-orig-size=\"290,370\" 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=\"Tri\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag63-20.png\" class=\"alignright size-medium wp-image-14272\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag63-20-235x300.png\" alt=\"\" width=\"235\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag63-20-235x300.png 235w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag63-20.png 290w\" sizes=\"auto, (max-width: 235px) 100vw, 235px\" \/><\/a>Seja [<em>ABC<\/em>] um tri\u00e2ngulo tal que \\(\\overline {AC} = 4\\) e \\(\\overline {AB} = \\overline {BC} = 6\\).<\/p>\n<p>Seja <em>M<\/em> o ponto m\u00e9dio de [<em>AB<\/em>].<\/p>\n<\/blockquote>\n<ol>\n<li>Seja [<em>BN<\/em>] a altura do tri\u00e2ngulo relativa ao v\u00e9rtice <em>B<\/em>.<br \/>\nEnt\u00e3o, este segmento de reta \u00e9 perpendicular ao lado [<em>AC<\/em>] do tri\u00e2ngulo.<br \/>\nOra, como\u00a0\\(\\overline {AB} = \\overline {BC} \\), ent\u00e3o o ponto <em>B<\/em> pertence \u00e0 mediatriz de [<em>AC<\/em>].<br \/>\nComo a reta <em>BN<\/em> \u00e9 perpendicular a [<em>AC<\/em>] e cont\u00e9m um ponto (<em>B<\/em>) equidistante dos extremos desse segmento, ent\u00e3o essa reta <em>BN<\/em> \u00e9 a mediatriz de [<em>AC<\/em>].<br \/>\nFinalmente, como <em>N<\/em> \u00e9 um ponto de [<em>AC<\/em>] e da sua mediatriz, ent\u00e3o ser\u00e1 o ponto m\u00e9dio de [<em>AC<\/em>].<br \/>\n\u00ad<\/li>\n<li>Os segmentos de reta [<em>BN<\/em>] e [<em>CM<\/em>] s\u00e3o duas medianas do tri\u00e2ngulo [<em>ABC<\/em>], que se intersetam no ponto <em>Q<\/em>. Por isso, <em>Q<\/em> \u00e9 o baricentro do tri\u00e2ngulo [<em>ABC<\/em>].<br \/>\n\u00ad<\/li>\n<li>Aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [<em>BCN<\/em>], temos:\\[\\overline {BN} = \\sqrt {{6^2} &#8211; {2^2}} = \\sqrt {32} = 4\\sqrt 2 \\]<br \/>\nComo o baricentro divide cada uma das medianas em dois segmentos com comprimentos de raz\u00e3o 2:1, vem: \\[\\overline {QN} = \\frac{1}{2}\\overline {BQ} = \\frac{1}{3}\\overline {BN} = \\frac{1}{3} \\times \\sqrt {32} = \\frac{{\\sqrt {32} }}{3} = \\frac{{4\\sqrt 2 }}{3}\\]<br \/>\n\u00ad<\/li>\n<li>No tri\u00e2ngulo ret\u00e2ngulo [<em>CNQ<\/em>], vem:<br \/>\n\\[\\begin{array}{*{20}{l}}{{\\mathop{\\rm tg}\\nolimits} A\\widehat CM = \\frac{{\\overline {QN} }}{{\\overline {NC} }}}&amp; \\Leftrightarrow &amp;{{\\mathop{\\rm tg}\\nolimits} A\\widehat CM = \\frac{{\\frac{{4\\sqrt 2 }}{3}}}{2}}\\\\{}&amp; \\Leftrightarrow &amp;{{\\mathop{\\rm tg}\\nolimits} A\\widehat CM = \\frac{{2\\sqrt 2 }}{3}}\\\\{}&amp; \\Leftrightarrow &amp;{A\\widehat CM = {{{\\mathop{\\rm tg}\\nolimits} }^{ &#8211; 1}}(\\frac{{2\\sqrt 2 }}{3})}\\\\{}&amp;{}&amp;{A\\widehat CM \\approx 43,3^\\circ }\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14267' onClick='GTTabs_show(0,14267)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Seja [ABC] um tri\u00e2ngulo tal que \\(\\overline {AC} = 4\\) e \\(\\overline {AB} = \\overline {BC} = 6\\). Seja M o ponto m\u00e9dio de [AB]. Determina a medida da amplitude do&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14273,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,489],"tags":[426,491,493,490,492],"series":[],"class_list":["post-14267","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-trigonometria-9--ano","tag-9-o-ano","tag-cosseno","tag-razao-trigonometrica","tag-seno","tag-tangente"],"views":1748,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag63-20a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14267","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=14267"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14267\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14273"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14267"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14267"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14267"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14267"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}