{"id":141997,"date":"2019-11-20T16:47:47","date_gmt":"2019-11-20T18:47:47","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=141997"},"modified":"2024-06-23T06:34:01","modified_gmt":"2024-06-23T09:34:01","slug":"problema-para-ajudar-na-escola-uma-afirmacao","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-uma-afirmacao\/","title":{"rendered":".Problema para ajudar na escola: Uma afirma\u00e7\u00e3o"},"content":{"rendered":"<p class=\"g2\" ><strong><a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=220638\" rel=\"noopener\" target=\"_blank\">&#9999; <span style=\"color:#A0522D;font-size:14px;\">Link do problema para dispositivos da <em>Apple<\/em>.<\/span><\/a><\/strong><\/p>\n<div class=\"g1n\" style=\"line-height:25px;\">\n<p class=\"tit\" style=\"line-height:20px; margin-bottom: 10px;\">Problema<br \/>\n<span style=\"font-size:12px;color:#00BF00\"> (A partir do 9\u00ba ano do E. F. &#8211; <span style=\"font-weight: normal;\"> N\u00edvel de dificuldade: Dif\u00edcil<\/span>)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p>A circunfer\u00eancia [tex]~\\textcolor{red}{c}~[\/tex] de centro [tex] \\textcolor{red}{~O~}[\/tex], que aparece na figura, \u00e9 tangente \u00e0s semirretas [tex] \\textcolor{#0099FF}{PX}[\/tex] e [tex] \\textcolor{#0099FF}{PY}[\/tex] nos pontos [tex] A[\/tex]  e [tex] B[\/tex], respectivamente.<\/p>\n<ul>\n<li><strong>Afirma\u00e7\u00e3o:<\/strong> O centro [tex] \\textcolor{red}{~O~}[\/tex] pertence \u00e0 bissetriz interna do \u00e2ngulo [tex] \\textcolor{#0099FF}{X\\hat{P}Y}.[\/tex] <\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/09\/prob382.png\" style=\"border: none;\"  alt=\"\" width=\"393\" height=\"339\" class=\"aligncenter size-full wp-image-136621\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/09\/prob382.png 393w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/09\/prob382-300x259.png 300w\" sizes=\"auto, (max-width: 393px) 100vw, 393px\" \/><\/p>\n<p><strong>Essa afirma\u00e7\u00e3o \u00e9 verdadeira?<\/strong> Justifique sua resposta.<\/p>\n<p style=\"margin:10px\">\n<\/div>\n<div class=\"cb36 g1\"style=\"margin:40px; width:80%;margin-left:auto; margin-right:auto\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" style=\"border:none\"  alt=\"explicador_p\" width=\"80\" height=\"96\" class=\"alignleft size-full wp-image-1001\" \/> <\/p>\n<p class=\"c3 tit\" style=\"line-height:20px; margin-bottom: 10px;\"> <strong>Lembretes:<\/strong> <\/p>\n<p><span class=\"c3\"> <strong>(1)<\/strong> <\/span> Toda tangente a uma circunfer\u00eancia \u00e9 perpendicular ao raio no ponto de tang\u00eancia.<br \/>\n<span class=\"c3\"> <strong>(2)<\/strong> <\/span> <strong>Congru\u00eancia de tri\u00e2ngulos ret\u00e2ngulos:<\/strong> Se dois tri\u00e2ngulos ret\u00e2ngulos t\u00eam ordenadamente congruentes um cateto e a hipotenusa, ent\u00e3o estes tri\u00e2ngulos s\u00e3o congruentes.<br \/>\n<span class=\"c3\" style=\"font-size: 25px;\"> &#10000; <\/span><strong>Nota\u00e7\u00e3o:<\/strong><\/span> Denotaremos o segmento definido por dois pontos, digamos [tex]E[\/tex] e [tex]F[\/tex],  por [tex]\\overline {EF} [\/tex] e o seu comprimento por [tex]EF.[\/tex]\n<\/div>\n<div class=\"g1n\" style=\"line-height:30px;\">\n<p class=\"tit\" style=\"line-height:30px; margin-bottom: 10px;\">Solu\u00e7\u00e3o <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nA partir dos dados do problema, devemos garantir que a semirreta [tex]PO[\/tex] \u00e9 a bissetriz do \u00e2ngulo [tex] \\textcolor{#0099FF}{X\\hat{P}Y}.[\/tex]<br \/>\nPara isso, inicialmente vamos tra\u00e7ar os segmentos [tex]\\overline {OA}\\, [\/tex] e  [tex]\\, \\overline {OB}.[\/tex]<br \/>\nDe acordo com o <span class=\"c3\"> <strong>Lembrete (1)<\/strong> <\/span>, definimos dois tri\u00e2ngulos ret\u00e2ngulos: o tri\u00e2ngulo [tex]PAO[\/tex] e o tri\u00e2ngulo [tex]PBO.[\/tex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/11\/prob382_1.png\" style=\"border: none;\"  alt=\"\" width=\"452\" height=\"403\" class=\"aligncenter size-full wp-image-142009\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/11\/prob382_1.png 452w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/11\/prob382_1-300x267.png 300w\" sizes=\"auto, (max-width: 452px) 100vw, 452px\" \/><\/p>\n<p>Observe que os segmentos [tex]\\overline {OA}\\, [\/tex] e  [tex]\\, \\overline {OB}[\/tex] s\u00e3o raios de [tex]~\\textcolor{red}{c}~[\/tex] e, portanto, s\u00e3o congruentes. Assim, pelo <span class=\"c3\"> <strong>Lembrete (2)<\/strong><\/span>, os tri\u00e2ngulos [tex]PAO[\/tex] e  [tex]PBO[\/tex] s\u00e3o congruentes.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/12\/prob382_4.png\" style=\"border: none; \"alt=\"\" width=\"450\" height=\"397\" class=\"aligncenter size-full wp-image-143852\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/12\/prob382_4.png 450w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/12\/prob382_4-300x265.png 300w\" sizes=\"auto, (max-width: 450px) 100vw, 450px\" \/><\/p>\n<p>Com isso, particularmente, os \u00e2ngulos [tex]\\textcolor{#FF33CC}{A\\hat{P}O}[\/tex] e [tex]\\textcolor{#006400}{B\\hat{P}O}[\/tex] t\u00eam a mesma medida e, portanto, a semirreta [tex]PO[\/tex] \u00e9 a bissetriz interna do tri\u00e2ngulo [tex] \\textcolor{#0099FF}{X\\hat{P}Y}.[\/tex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/11\/prob382_3.png\" style=\"border: none; \"alt=\"\" width=\"463\" height=\"404\" class=\"aligncenter size-full wp-image-142023\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/11\/prob382_3.png 463w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/11\/prob382_3-300x262.png 300w\" sizes=\"auto, (max-width: 463px) 100vw, 463px\" \/><\/p>\n<p>Logo, o centro [tex] \\textcolor{red}{~O~}[\/tex] pertence \u00e0 bissetriz interna do \u00e2ngulo [tex] \\textcolor{#0099FF}{X\\hat{P}Y}[\/tex] e <strong>a afirma\u00e7\u00e3o \u00e9 verdadeira<\/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:20px\">\n<table style=\" height:auto; margin-left: auto; margin-right: auto;width:640px;\">\n<tr>\n<td class=\"bf\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1001 alignright\" style=\"border: 0px;\" alt=\"\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" width=\"80\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:520px;background-color: #FFF5EE; padding:10px; font-family: calibri; font-size: 13px;color: #1C1C1C; line-height: 20px; \"> Se for conveniente, voc\u00ea pode obter um arquivo PDF desta p\u00e1gina, com o problema e a solu\u00e7\u00e3o, clicando no bot\u00e3o abaixo.<br \/>\nVoc\u00ea pode abrir o arquivo diretamente no seu navegador (<em>Chrome<\/em>, <em>Edge<\/em>, <em>Firefox<\/em>, <em>Safari<\/em>, entre outros), mas tamb\u00e9m pode utilizar o software gratuito <em><strong>Adobe Acrobat Reader<\/strong><\/em>.<br \/>\nCaso o dispositivo que voc\u00ea est\u00e1 utilizando n\u00e3o tenha o <strong><em>Acrobat Reader<\/em><\/strong> instalado, \u00e9 s\u00f3 clicar <a href=\"http:\/\/get.adobe.com\/br\/reader\/\" rel=\"noopener\" target=\"_blank\"><strong>AQUI<\/strong><\/a> para fazer o download adequado ao seu dispositivo.<\/p>\n<\/td>\n<\/tr>\n<\/table>\n<div style=\" float:right;  width: 100px;\">\n<p style=\"text-align: center; border-style: dotted; border-color :#FFFAFA; border-width :thin; width:90px;  height:40px;  padding: 5px; background-color: #800000;\">\n<a href=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/12\/Uma-afirma\u00e7\u00e3o.pdf\" title=\" Arquivo PDF\" download=\"Clubes_ Uma afirma\u00e7\u00e3o.pdf\"><span style=\"color: #FFFAFA; font-family: georgia; font-size: 16px;\"><strong>Arquivo <\/strong><\/span><\/a>\n <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>&#9999; Link do problema para dispositivos da Apple. Problema (A partir do 9\u00ba ano do E. F. &#8211; N\u00edvel de dificuldade: Dif\u00edcil) A circunfer\u00eancia [tex]~\\textcolor{red}{c}~[\/tex] de centro [tex] \\textcolor{red}{~O~}[\/tex], que aparece na figura, \u00e9 tangente \u00e0s semirretas [tex] \\textcolor{#0099FF}{PX}[\/tex] e [tex] \\textcolor{#0099FF}{PY}[\/tex] nos pontos [tex] A[\/tex] e [tex] B[\/tex], respectivamente. Afirma\u00e7\u00e3o: O centro [tex] \\textcolor{red}{~O~}[\/tex] &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-uma-afirmacao\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-141997","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/141997","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\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=141997"}],"version-history":[{"count":35,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/141997\/revisions"}],"predecessor-version":[{"id":220640,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/141997\/revisions\/220640"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=141997"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}