{"id":84661,"date":"2017-09-20T14:03:12","date_gmt":"2017-09-20T17:03:12","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=84661"},"modified":"2023-04-08T02:46:29","modified_gmt":"2023-04-08T05:46:29","slug":"problemao-tetraedro-regular","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-tetraedro-regular\/","title":{"rendered":".Problem\u00e3o:  Tetraedro regular"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1\" style=\"line-height:25px;\">\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: 20px;border: 0 none\"\/>\nDado o tetraedro regular [tex]ABCD[\/tex] com aresta medindo [tex]a[\/tex], considere [tex]E[\/tex] e [tex]F[\/tex] pontos m\u00e9dios das arestas [tex]\\overline{AC}[\/tex] e [tex]\\overline{BD}[\/tex], respectivamente.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2017\/02\/Problemao-Figura-1.png\" style=\"border:none\" alt=\"Problemao Figura 1\" width=\"390\" height=\"360\" class=\"aligncenter size-full wp-image-71617\" \/><\/p>\n<p><strong>Determine a dist\u00e2ncia entre os pontos [tex]E[\/tex] e [tex]F[\/tex].<\/strong><\/p>\n<p style=\"margin:10px\">\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1n\" style=\"line-height:30px;\">\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\"\/>\nObserve o tetraedro abaixo, no qual tra\u00e7amos os segmentos [tex]\\overline{AF}[\/tex] e [tex]\\overline{CF}[\/tex], bem como o segmento [tex]\\overline{EF}[\/tex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2017\/09\/Rob_11.png\" style=\"border:none\" alt=\"\" width=\"519\" height=\"461\" class=\"aligncenter size-full wp-image-84664\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2017\/09\/Rob_11.png 519w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2017\/09\/Rob_11-300x266.png 300w\" sizes=\"auto, (max-width: 519px) 100vw, 519px\" \/><\/p>\n<ul>\n<li>Como o tetraedro \u00e9 regular, suas faces s\u00e3o tri\u00e2ngulos equil\u00e1teros; assim, na face [tex]BCD[\/tex] o segmento [tex]\\overline{CF}[\/tex] \u00e9 mediana e altura (e tamb\u00e9m bissetriz e mediatriz) e, com isso, [tex]BCF[\/tex] e [tex]CFD[\/tex] s\u00e3o tri\u00e2ngulos ret\u00e2ngulos. Portanto, temos que [tex]FD=\\dfrac{a}{2}[\/tex] e [tex]\\boxed{CF=\\dfrac{a\\sqrt{3}}{2}}[\/tex].<\/li>\n<li>De maneira an\u00e1loga, se observarmos a face [tex]ABD[\/tex], podemos concluir que o segmento [tex]\\overline{AF}[\/tex] \u00e9 tamb\u00e9m uma mediana e altura e, ent\u00e3o,  [tex]\\boxed{AF=\\dfrac{a\\sqrt{3}}{2}}[\/tex].<\/li>\n<p style=\"text-align:center\";><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2017\/09\/rob23.png\"  alt=\"\" width=\"249\" height=\"221\" class=\"alignnone size-full wp-image-84673\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2017\/09\/rob24.png\" alt=\"\" width=\"247\" height=\"219\" class=\"alignnone size-full wp-image-84692\" \/><\/p>\n<p style=\"margin:20px\">\n<li>Note, agora, que [tex]AFC[\/tex] \u00e9 um tri\u00e2ngulo is\u00f3sceles, com [tex]AF=CF[\/tex]; dessa forma, [tex]\\overline{FE}[\/tex] \u00e9 mediana e altura com rela\u00e7\u00e3o ao lado [tex]\\overline{AC}[\/tex].<br \/>\nAplicando o Teorema de Pit\u00e1goras ao tri\u00e2ngulo [tex]FEC[\/tex], segue, finalmente,  que:<br \/>\n[tex]\\qquad (EF)^2+(EC)^2=(CF)^2[\/tex]<br \/>\n[tex]\\qquad (EF)^2=(CF)^2-(EC)^2[\/tex]<br \/>\n[tex]\\qquad (EF)^2=\\left(\\dfrac{a\\sqrt{3}}{2}\\right)^2-\\left(\\dfrac{a}{2}\\right)^2[\/tex]<br \/>\n[tex]\\qquad (EF)^2=\\dfrac{2a^2}{4}[\/tex]<br \/>\n[tex]\\qquad EF=\\dfrac{a\\sqrt{2}}{2}[\/tex].<\/li>\n<\/ul>\n<p>Logo, a dist\u00e2ncia entre os pontos [tex]E[\/tex] e [tex]F[\/tex] \u00e9 [tex] \\, \\fcolorbox{black}{#eee0e5}{$\\dfrac{a\\sqrt{2}}{2}$} \\, [\/tex] unidades de comprimento.<\/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:10px\">\n","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 3\u00ba ano do E. M.) Dado o tetraedro regular [tex]ABCD[\/tex] com aresta medindo [tex]a[\/tex], considere [tex]E[\/tex] e [tex]F[\/tex] pontos m\u00e9dios das arestas [tex]\\overline{AC}[\/tex] e [tex]\\overline{BD}[\/tex], respectivamente. Determine a dist\u00e2ncia entre os pontos [tex]E[\/tex] e [tex]F[\/tex]. Solu\u00e7\u00e3o Observe o tetraedro abaixo, no qual tra\u00e7amos os segmentos [tex]\\overline{AF}[\/tex] e [tex]\\overline{CF}[\/tex], bem como &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problemao-tetraedro-regular\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":164,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-84661","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/84661","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\/164"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=84661"}],"version-history":[{"count":42,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/84661\/revisions"}],"predecessor-version":[{"id":186558,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/84661\/revisions\/186558"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=84661"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}