{"id":8082,"date":"2012-04-09T01:24:16","date_gmt":"2012-04-09T00:24:16","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8082"},"modified":"2022-01-30T19:45:11","modified_gmt":"2022-01-30T19:45:11","slug":"um-triangulo-equilatero-3","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8082","title":{"rendered":"Um tri\u00e2ngulo equil\u00e1tero"},"content":{"rendered":"<p><ul id='GTTabs_ul_8082' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8082' class='GTTabs_curr'><a  id=\"8082_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8082' ><a  id=\"8082_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_8082'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag224-81.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8085\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8085\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag224-81.jpg\" data-orig-size=\"205,189\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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;}\" data-image-title=\"Tri\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag224-81.jpg\" class=\"alignright wp-image-8085 size-full\" title=\"Tri\u00e2ngulo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag224-81.jpg\" alt=\"\" width=\"205\" height=\"189\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag224-81.jpg 205w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag224-81-150x138.jpg 150w\" sizes=\"auto, (max-width: 205px) 100vw, 205px\" \/><\/a>Considere o tri\u00e2ngulo ret\u00e2ngulo [ABC] de lado $a$.<\/p>\n<p>Inscreve-se nesse tri\u00e2ngulo um ret\u00e2ngulo [MNPQ].<\/p>\n<p>Fa\u00e7a-se $\\overline {AM}\u00a0 = x$.<\/p>\n<p>Para que valor de $x$ a \u00e1rea do ret\u00e2ngulo \u00e9 m\u00e1xima?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8082' onClick='GTTabs_show(1,8082)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8082'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag224-81.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8085\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8085\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag224-81.jpg\" data-orig-size=\"205,189\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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;}\" data-image-title=\"Tri\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag224-81.jpg\" class=\"alignright wp-image-8085 size-full\" title=\"Tri\u00e2ngulo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag224-81.jpg\" alt=\"\" width=\"205\" height=\"189\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag224-81.jpg 205w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag224-81-150x138.jpg 150w\" sizes=\"auto, (max-width: 205px) 100vw, 205px\" \/><\/a>Como o tri\u00e2ngulo\u00a0 [ABC] \u00e9 equi\u00e2ngulo, temos: $$\\operatorname{tg} 60^\\circ\u00a0 = \\frac{{\\overline {QM} }}{{\\overline {AM} }}$$ donde $$\\operatorname{tg} 60^\\circ\u00a0 = \\frac{{\\overline {QM} }}{x} \\Leftrightarrow QM = \\sqrt 3 \\,x$$<\/p>\n<p>A \u00e1rea do ret\u00e2ngulo pode ser expressa por: $$A(x) = \\left( {a &#8211; 2x} \\right) \\times \\sqrt 3 x$$ com $0 &lt; x &lt; \\frac{a}{2}$.<\/p>\n<p>Ora,\u00a0$$A'(x) =\u00a0 &#8211; 2\\sqrt 3 x + \\sqrt 3 \\left( {a &#8211; 2x} \\right) = \\sqrt 3 a &#8211; 4\\sqrt 3 x$$<\/p>\n<p>Como $A'(x) = 0 \\Leftrightarrow \\sqrt 3 a &#8211; 4\\sqrt 3 x = 0 \\Leftrightarrow x = \\frac{a}{4}$, temos:<\/p>\n<table class=\" aligncenter\" style=\"width: 80%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$x$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$\\frac{a}{4}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$\\frac{a}{2}$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">Sinal de $A'(x) = \\sqrt 3 a &#8211; 4\\sqrt 3 x$<\/td>\n<td style=\"text-align: center; background-color: #a9a9a9; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-$<\/td>\n<td style=\"text-align: center; background-color: #a9a9a9; border: #00008b 1px solid;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">Varia\u00e7\u00e3o de $A$<\/td>\n<td style=\"text-align: center; background-color: #a9a9a9; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\nearrow $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$\\frac{{\\sqrt 3 }}{8}{a^2}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\searrow $<\/td>\n<td style=\"text-align: center; background-color: #a9a9a9; border: #00008b 1px solid;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>$$A(\\frac{a}{4}) = \\left( {a &#8211; \\frac{a}{2}} \\right) \\times \\sqrt 3 \\frac{a}{4} = \\frac{{\\sqrt 3 }}{8}{a^2}$$<\/p>\n<p>A\u00a0\u00e1rea \u00e9 m\u00e1xima para $x = \\frac{a}{4}$.<br \/>\n\u00ad<\/p>\n<p style=\"text-align: center;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":522,\r\n\"height\":485,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 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