{"id":14326,"date":"2018-04-09T22:14:25","date_gmt":"2018-04-09T21:14:25","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14326"},"modified":"2022-01-11T14:32:55","modified_gmt":"2022-01-11T14:32:55","slug":"um-triangulo-escaleno-e-retangulo","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14326","title":{"rendered":"Um tri\u00e2ngulo escaleno e ret\u00e2ngulo"},"content":{"rendered":"<p><ul id='GTTabs_ul_14326' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14326' class='GTTabs_curr'><a  id=\"14326_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14326' ><a  id=\"14326_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_14326'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag69-8.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14327\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14327\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag69-8.png\" data-orig-size=\"505,393\" 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\/9V2Pag69-8.png\" class=\"alignright size-medium wp-image-14327\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag69-8-300x233.png\" alt=\"\" width=\"300\" height=\"233\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag69-8-300x233.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag69-8.png 505w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Relativamente \u00e0 figura, que n\u00e3o est\u00e1 desenhada \u00e0 escala, sabe-se que:<\/p>\n<ul>\n<li>o tri\u00e2ngulo [<em>ABC<\/em>] \u00e9 escaleno e \u00e9 ret\u00e2ngulo em <em>B<\/em>;<\/li>\n<li>os pontos <em>E<\/em> e <em>P<\/em> pertencem ao segmento de reta [<em>AC<\/em>];<\/li>\n<li>o ponto <em>D<\/em> pertence ao segmento de reta [<em>AB<\/em>];<\/li>\n<li>o tri\u00e2ngulo [<em>ADE<\/em>] \u00e9 ret\u00e2ngulo em <em>D<\/em>;<\/li>\n<li>o ponto <em>Q<\/em> pertence ao segmento de reta [<em>BC<\/em>];<\/li>\n<li><em>PCQ<\/em> \u00e9 um arco de circunfer\u00eancia;<\/li>\n<\/ul>\n<ol>\n<li>Admite que \\(\\overline {AD} = 20\\),\u00a0\\(\\overline {AE} = 25\\) e\u00a0\\(\\overline {AC} = 40\\).<br \/>\nDetermina \\(\\overline {BC} \\).<br \/>\nMostra como chegaste \u00e0 tua resposta.<\/li>\n<li>Admite agora que a amplitude do \u00e2ngulo <em>DAE<\/em> \u00e9 37 graus.<br \/>\nDetermina a amplitude, em graus, do arco <em>PCQ<\/em>.<br \/>\nMostra como chegaste \u00e0 tua resposta.<\/li>\n<li>Qual as afirma\u00e7\u00f5es seguintes \u00e9 verdadeira?<br \/>\n[A]\u00a0\\({\\mathop{\\rm sen}\\nolimits} A\\widehat CB = \\frac{{\\overline {BC} }}{{\\overline {AC} }}\\)<br \/>\n[B]\u00a0\\({\\mathop{\\rm sen}\\nolimits} A\\widehat CB = \\frac{{\\overline {AC} }}{{\\overline {BC} }}\\)<br \/>\n[C]\u00a0\\(\\cos A\\widehat CB = \\frac{{\\overline {BC} }}{{\\overline {AC} }}\\)<br \/>\n[D]\u00a0\\(\\cos A\\widehat CB = \\frac{{\\overline {AC} }}{{\\overline {BC} }}\\)<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14326' onClick='GTTabs_show(1,14326)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14326'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag69-8.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14327\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14327\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag69-8.png\" data-orig-size=\"505,393\" 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\/9V2Pag69-8.png\" class=\"alignright size-medium wp-image-14327\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag69-8-300x233.png\" alt=\"\" width=\"300\" height=\"233\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag69-8-300x233.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag69-8.png 505w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [<em>ADE<\/em>], vem:<br \/>\n\\(\\overline {DE} = \\sqrt {{{\\overline {AE} }^2} &#8211; {{\\overline {AD} }^2}} = \\sqrt {{{25}^2} &#8211; {{20}^2}} = \\sqrt {225} = 15\\)<br \/>\nTendo em considera\u00e7\u00e3o a semelhan\u00e7a dos tri\u00e2ngulos [<em>ADE<\/em>] e [<em>ABC<\/em>], temos:<br \/>\n\\[\\begin{array}{*{20}{c}}{\\frac{{\\overline {BC} }}{{\\overline {DE} }} = \\frac{{\\overline {AC} }}{{\\overline {AE} }}}&amp; \\Leftrightarrow &amp;{\\frac{{\\overline {BC} }}{{15}} = \\frac{{40}}{{25}}}&amp; \\Leftrightarrow &amp;{\\overline {BC} = \\frac{{15 \\times 40}}{{25}}}&amp; \\Leftrightarrow &amp;{\\overline {BC} = 24}\\end{array}\\]<\/li>\n<li>No tri\u00e2ngulo ret\u00e2ngulo [<em>ABC<\/em>], os \u00e2ngulos <em>BAC<\/em> e <em>ACB<\/em> s\u00e3o complementares. Logo,\u00a0\\(A\\widehat CB = 90^\\circ &#8211; B\\widehat AC = 90^\\circ &#8211; 37^\\circ = 53^\\circ \\).<br \/>\nTendo em considera\u00e7\u00e3o que <em>ACB<\/em> \u00e9 um \u00e2ngulo inscrito, vem: \\(\\overparen{PCQ} = 360^\\circ &#8211; \\overparen{PQ} = 360^\\circ &#8211; 2 \\times A\\widehat CB = 360^\\circ &#8211; 2 \\times 53^\\circ = 254^\\circ \\).<br \/>\n\u00ad<\/li>\n<li>A afirma\u00e7\u00e3o verdadeira \u00e9 a [C].<br \/>\n[C]\u00a0\\(\\cos A\\widehat CB = \\frac{{\\overline {BC} }}{{\\overline {AC} }}\\)<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14326' onClick='GTTabs_show(0,14326)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Relativamente \u00e0 figura, que n\u00e3o est\u00e1 desenhada \u00e0 escala, sabe-se que: o tri\u00e2ngulo [ABC] \u00e9 escaleno e \u00e9 ret\u00e2ngulo em B; os pontos E e P pertencem ao segmento de reta&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14329,"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-14326","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":2894,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag69-8a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14326","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=14326"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14326\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14329"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14326"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14326"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14326"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14326"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}