{"id":33194,"date":"2015-03-20T11:36:07","date_gmt":"2015-03-20T14:36:07","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=33194"},"modified":"2023-03-21T21:22:09","modified_gmt":"2023-03-22T00:22:09","slug":"problemao-um-angulo-em-um-tetraedro","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-um-angulo-em-um-tetraedro\/","title":{"rendered":".Problem\u00e3o:  Um \u00e2ngulo em um tetraedro"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1n\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Problema<br \/>\n<span style=\"font-size:12px;color:#800000\"> (Indicado a partir do 3\u00ba ano do E. M.)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 10px;border: 0 none\"\/>\n<p style=\"margin:20px\">\n<p>Seja [tex]P[\/tex] um ponto pertencente \u00e0 aresta [tex]\\overline{AD}[\/tex] do tetraedro regular [tex]ABCD[\/tex], como representado abaixo.<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2022\/07\/f235.png\"  style=\"border: none;\" alt=\"\" width=\"341\" height=\"324\" class=\"aligncenter size-full wp-image-181253\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2022\/07\/f235.png 341w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2022\/07\/f235-300x285.png 300w\" sizes=\"auto, (max-width: 341px) 100vw, 341px\" \/><br \/>\nSe [tex]P[\/tex] \u201cdesliza\u201d sobre [tex]\\overline{AD}[\/tex], teremos o tri\u00e2ngulo [tex]\\triangle BPC[\/tex] variando com [tex]P[\/tex] e, consequentemente, o \u00e2ngulo [tex]B\\widehat{P}C[\/tex] tamb\u00e9m variar\u00e1.<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2022\/07\/f235_2.png\"  style=\"border: none;\" alt=\"\" width=\"282\" height=\"302\" class=\"aligncenter size-full wp-image-181255\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2022\/07\/f235_2.png 282w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2022\/07\/f235_2-280x300.png 280w\" sizes=\"auto, (max-width: 282px) 100vw, 282px\" \/><br \/>\n<strong>a)<\/strong> Verifique, justificando sua resposta, que o tri\u00e2ngulo [tex]BPC[\/tex] \u00e9 sempre is\u00f3sceles.<br \/>\n<strong>b)<\/strong> Determine a posi\u00e7\u00e3o de [tex]P[\/tex] para que a medida [tex]\\beta[\/tex] seja m\u00e1xima. Justifique sua resposta.<\/p>\n<p style=\"margin:10px\">\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1n\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Solu\u00e7\u00e3o <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<strong>a)<\/strong> Observe, inicialmente, que:<\/p>\n<ul>\n<li>[tex]\\overline{CD}\\equiv \\overline{BD}[\/tex] (s\u00e3o arestas do tetraedro)\n<\/li>\n<li>[tex]C\\widehat{D}P \\equiv B\\widehat{D}P[\/tex] (a medida dos \u00e2ngulos \u00e9 [tex]60^\\circ[\/tex])\n<\/li>\n<li>o lado [tex]\\overline{DP}[\/tex] \u00e9 comum aos tri\u00e2ngulos [tex]\\triangle CDP[\/tex] e [tex]\\triangle BDP[\/tex].<\/li>\n<\/ul>\n<p>Logo, pelo crit\u00e9rio de congru\u00eancia <strong>LAL<\/strong> (lado, \u00e2ngulo, lado), temos que [tex]\\triangle CDP \\equiv \\triangle BDP[\/tex] e, portanto, [tex]\\overline{CP} \\equiv \\overline{BP}[\/tex].<br \/>\n <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2022\/07\/f235_3.png\"  style=\"border: none;\" alt=\"\" width=\"304\" height=\"312\" class=\"aligncenter size-full wp-image-181256\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2022\/07\/f235_3.png 304w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2022\/07\/f235_3-292x300.png 292w\" sizes=\"auto, (max-width: 304px) 100vw, 304px\" \/><br \/>\nAssim, o tri\u00e2ngulo [tex]\\triangle BPC[\/tex] \u00e9 is\u00f3sceles.<\/p>\n<p style=\"margin:10px\">\n<p><strong>b)<\/strong> Observe que em um tri\u00e2ngulo is\u00f3sceles, se a base \u00e9 fixa, quanto menor for a medida dos lados congruentes, maior ser\u00e1 a medida do \u00e2ngulo oposto \u00e0 base. (Voc\u00ea pode visualizar essa <strong>propriedade <\/strong>no applet abaixo.)<br \/>\nAssim, no tri\u00e2ngulo em quest\u00e3o, a menor medida poss\u00edvel para os lados do tri\u00e2ngulo is\u00f3sceles \u00e9 quando estes coincidem com as alturas das faces \u00e0s quais pertencem.<br \/>\n<strong>Portanto, a medida [tex]\\beta[\/tex]  do \u00e2ngulo [tex]B\\widehat{P}C[\/tex] ser\u00e1 m\u00e1xima quando [tex]P[\/tex] for exatamente o ponto m\u00e9dio da aresta [tex]\\overline{AD}.[\/tex]<br \/>\n<\/strong><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\" style=\"line-height:20px;\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Visualiza\u00e7\u00e3o da propriedade<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 10px;border: 0 none\"\/>\n<p style=\"margin:20px\">\n<p>Para visualizar a propriedade, utilize o <em>applet <\/em>abaixo.<br \/>\nVoc\u00ea tamb\u00e9m tem a op\u00e7\u00e3o de copiar o arquivo <strong>.ggb <\/strong> disponibilizado abaixo e utilizar o <em>applet<\/em> no GeoGebra do seu computador.<\/p>\n<div style=\"text-align:center;\">\n<input  value=\"Mostrar o Applet\" style=\"padding: 0px; width: 180px; height: 40px; background-color:#9c3838; font-size: 20px; font-family: Calibri, sans-serif; font-weight: bold; color: #FFFAFA;\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar o Applet';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Mostrar o Applet';}\"    type=\"button\" \/>    <\/p>\n<div style=\"display: none;\">\n<div style=\"border: 1px dotted #800000; padding: 10px;padding-right: 15px; padding-left: 15px;  width: 90%; margin-left:auto; margin-right:auto\">\n<p style=\"text-align: justify;\">\n\t1) Abra o <em>applet<\/em> e aguarde o arquivo carregar.<\/p>\n<p style=\"margin:20px\">\n<p>\tPara abrir o <em>applet<\/em>, clique <a href=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/rBLglPlP\/width\/727\/height\/694\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/true\/ld\/false\/sri\/false\/at\/auto\" target=\"ap1\"><span style=\"color: #800000;font-size: 16px;\"><strong>AQUI<\/strong><\/span><\/a><br \/>\n<iframe  name=\"ap1\"  width=\"727\" height=\"694\" style=\"border: 4px solid #800000;\"> <\/iframe><span style=\"font-size: 14px;\"><br \/>\n\tOBMEP_ srdg, criado com o <a href=\"http:\/\/www.geogebra.org\/\" target=\"_blank\">GeoGebra<\/a><\/span><br \/>\n\t<span style=\"font-family: Georgia, Times, 'Times New Roman', serif; font-size: 13px; text-align: justify; color: #800000; line-height: 15px;\">Observamos que a planilha ajuda na visualiza\u00e7\u00e3o do resultado;<br \/>\nmas, matematicamente, n\u00e3o substitui sua demonstra\u00e7\u00e3o.<\/span>\n\t<\/div>\n<\/div>\n<\/div>\n<div>\n<p style=\"margin:40px\">\n<p style=\"text-align: center; border-style: solid;border-color:#FFFAFA; border-width :thin; width:90px;  height:40px;  padding: 5px; background-color: #9c3838;\">\n<a href=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/03\/triangulo1.ggb\" download=\"triangulo1.ggb\"><span style=\"font-family: Georgia, Times, 'Times New Roman', serif; font-size: 16px;color: #FFFAFA;\"><strong>Arquivo<\/strong><\/span><\/a><\/p>\n<\/div>\n<p style=\"margin:10px\">\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g3\">Participou da discuss\u00e3o o Clube: <strong>Fermatianos<\/strong>.<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 3\u00ba ano do E. M.) Seja [tex]P[\/tex] um ponto pertencente \u00e0 aresta [tex]\\overline{AD}[\/tex] do tetraedro regular [tex]ABCD[\/tex], como representado abaixo. Se [tex]P[\/tex] \u201cdesliza\u201d sobre [tex]\\overline{AD}[\/tex], teremos o tri\u00e2ngulo [tex]\\triangle BPC[\/tex] variando com [tex]P[\/tex] e, consequentemente, o \u00e2ngulo [tex]B\\widehat{P}C[\/tex] tamb\u00e9m variar\u00e1. a) Verifique, justificando sua resposta, que o tri\u00e2ngulo [tex]BPC[\/tex] \u00e9 &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problemao-um-angulo-em-um-tetraedro\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":165,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-33194","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/33194","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/165"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=33194"}],"version-history":[{"count":40,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/33194\/revisions"}],"predecessor-version":[{"id":185035,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/33194\/revisions\/185035"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=33194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}