{"id":14723,"date":"2018-04-20T22:06:09","date_gmt":"2018-04-20T21:06:09","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14723"},"modified":"2022-01-09T23:05:43","modified_gmt":"2022-01-09T23:05:43","slug":"uma-parabola-e-um-triangulo","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14723","title":{"rendered":"Uma par\u00e1bola e um tri\u00e2ngulo"},"content":{"rendered":"<p><ul id='GTTabs_ul_14723' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14723' class='GTTabs_curr'><a  id=\"14723_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14723' ><a  id=\"14723_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_14723'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-9.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14724\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14724\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-9.png\" data-orig-size=\"350,389\" 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=\"Gr\u00e1fico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-9.png\" class=\"alignright size-medium wp-image-14724\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-9-270x300.png\" alt=\"\" width=\"270\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-9-270x300.png 270w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-9.png 350w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/a>Na figura, est\u00e3o representados, num referencial cartesiano, parte do gr\u00e1fico de uma fun\u00e7\u00e3o quadr\u00e1tica <em>f<\/em> e o tri\u00e2ngulo [<em>OAB<\/em>].<br \/>\nSabe-se que:<\/p>\n<ul>\n<li>o ponto <em>O<\/em> \u00e9 a origem do referencial;<\/li>\n<li>o ponto <em>A<\/em> pertence ao gr\u00e1fico da fun\u00e7\u00e3o <em>f<\/em> e tem abcissa igual a 2;<\/li>\n<li>o ponto <em>B<\/em> pertence ao eixo das ordenadas;<\/li>\n<li>o tri\u00e2ngulo [<em>OAB<\/em>] \u00e9 ret\u00e2ngulo em <em>B<\/em>;<\/li>\n<li>a fun\u00e7\u00e3o <em>f<\/em> \u00e9 definida por \\(f\\left( x \\right) = a{x^2}\\), sendo <em>a<\/em> um n\u00famero positivo.<\/li>\n<\/ul>\n<ol>\n<li>Admite que a \u00e1rea do tri\u00e2ngulo [<em>OAB<\/em>] \u00e9 igual a 32.<br \/>\nDetermina o valor de <em>a<\/em>.<br \/>\nMostra como chegaste \u00e0 tua resposta.<\/li>\n<li>Admite agora que \\(f\\left( x \\right) = 3{x^2}\\).<br \/>\nResolve a equa\u00e7\u00e3o \\(f\\left( x \\right) = 5x &#8211; 2\\).<br \/>\nApresenta todos os c\u00e1lculos que efetuares.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14723' onClick='GTTabs_show(1,14723)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14723'>\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\/9V2Pag113-9.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14724\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14724\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-9.png\" data-orig-size=\"350,389\" 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=\"Gr\u00e1fico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-9.png\" class=\"alignright size-medium wp-image-14724\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-9-270x300.png\" alt=\"\" width=\"270\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-9-270x300.png 270w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-9.png 350w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/a>Ora,\u00a0\\({y_B} = {y_A} = f\\left( 2 \\right) = a \\times {2^2} = 4a\\).<br \/>\nComo\u00a0a \u00e1rea do tri\u00e2ngulo [<em>OAB<\/em>] \u00e9 igual a 32, vem:\\[\\begin{array}{*{20}{l}}{{A_{\\left[ {OAB} \\right]}} = 32}&amp; \\Leftrightarrow &amp;{\\frac{{\\overline {AB} \\times \\overline {OB} }}{2} = 32}\\\\{}&amp; \\Leftrightarrow &amp;{\\frac{{2 \\times 4a}}{2} = 32}\\\\{}&amp; \\Leftrightarrow &amp;{4a = 32}\\\\{}&amp; \\Leftrightarrow &amp;{a = 8}\\end{array}\\]<\/li>\n<li>Resolvendo a equa\u00e7\u00e3o, vem:<br \/>\n\\[\\begin{array}{*{20}{l}}{f\\left( x \\right) = 5x &#8211; 2}&amp; \\Leftrightarrow &amp;{3{x^2} = 5x &#8211; 2}\\\\{}&amp; \\Leftrightarrow &amp;{3{x^2} &#8211; 5x + 2 = 0}\\\\{}&amp; \\Leftrightarrow &amp;{x = \\frac{{5 \\mp \\sqrt {25 &#8211; 24} }}{6}}\\\\{}&amp; \\Leftrightarrow &amp;{x = \\frac{{5 \\mp 1}}{6}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = \\frac{2}{3}}&amp; \\vee &amp;{x = 1}\\end{array}}\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14723' onClick='GTTabs_show(0,14723)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura, est\u00e3o representados, num referencial cartesiano, parte do gr\u00e1fico de uma fun\u00e7\u00e3o quadr\u00e1tica f e o tri\u00e2ngulo [OAB]. Sabe-se que: o ponto O \u00e9 a origem do referencial; o ponto&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14726,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,249],"tags":[426,499,500],"series":[],"class_list":["post-14723","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-proporcionalidade-inversa-e-funcoes-algebricas","tag-9-o-ano","tag-funcao-quadratica","tag-parabola"],"views":4291,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-9a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14723","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=14723"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14723\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14726"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14723"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14723"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14723"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14723"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}