{"id":137743,"date":"2019-10-15T23:15:05","date_gmt":"2019-10-16T02:15:05","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=137743"},"modified":"2024-06-22T00:47:49","modified_gmt":"2024-06-22T03:47:49","slug":"problema-para-ajudar-na-escola-dois-poligonos","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-dois-poligonos\/","title":{"rendered":".Problema para ajudar na escola: Dois pol\u00edgonos"},"content":{"rendered":"<p class=\"g2\" ><strong><a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=220584\" 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:10px\">Problema<br \/>\n<span style=\"font-size:12px;color:#00BF00\"> (A partir do 8\u00ba ano do E. F. &#8211; <span style=\"font-weight: normal;\"> N\u00edvel de dificuldade: F\u00e1cil<\/span>)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p>(<strong>ONEM <\/strong> &#8211; 2010)  Os dois pol\u00edgonos mostrados na figura s\u00e3o regulares, t\u00eam um lado em comum e s\u00e3o coplanares.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/08\/prob362.png\" style=\"border: none;\" alt=\"\" width=\"400\" height=\"399\" class=\"aligncenter size-full wp-image-134539\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/08\/prob362.png 400w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/08\/prob362-150x150.png 150w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/08\/prob362-300x300.png 300w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/p>\n<p><strong>Determinar a medida angular [tex]\\textcolor{red}{\\alpha}[\/tex], em graus.<\/strong><\/p>\n<p style=\"margin:10px\">\n<\/div>\n<div class=\"cb36 g1n\" style=\"margin:40px;line-height:30px; width:75%; 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 ti3\" style=\"line-height:20px;margin:10px\"> Lembrete <\/p>\n<p style=\"margin:20px\">\n<p>A soma dos \u00e2ngulos internos de um pol\u00edgono convexo de [tex]n[\/tex] lados [tex](n\\gt 2)[\/tex] \u00e9 dada por:<br \/>\n[tex]\\qquad \\qquad \\boxed{S_{n}=\\left(n-2\\right)\\cdot 180^\\circ}.[\/tex]<\/p>\n<p style=\"margin:10px\">\n<\/div>\n<div class=\"g1n\" style=\"line-height:33px;\">\n<p class=\"tit\" style=\"line-height:20px;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<p>Observe que se conseguirmos obter as medidas angulares [tex]\\textcolor{#CC66FF}{\\theta}[\/tex] e [tex]\\textcolor{#339900}{\\beta}[\/tex] indicadas na figura abaixo, facilmente obtemos a medida [tex]\\textcolor{red}{\\alpha}[\/tex], uma vez que [tex]\\boxed{\\textcolor{red}{\\alpha}+\\textcolor{#339900}{\\beta}+\\textcolor{#CC66FF}{\\theta}=360^\\circ}.[\/tex]<br \/>\nVamos faz\u00ea-lo por etapas.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/10\/prob362_1.png\" style=\"border:none\"  alt=\"\" width=\"400\" height=\"398\" class=\"aligncenter size-full wp-image-137725\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/10\/prob362_1.png 400w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/10\/prob362_1-150x150.png 150w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/10\/prob362_1-300x300.png 300w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/p>\n<ul>\n<li>O pol\u00edgono colorido de verde \u00e9 um pent\u00e1gono.<br \/>\nPela f\u00f3rmula citada no <span class=\"c3\"><strong>Lembrete<\/strong><\/span>, a soma dos \u00e2ngulos internos de um pent\u00e1gono \u00e9<br \/>\n[tex]\\qquad \\qquad S_5=\\left(5-2\\right)\\cdot 180^\\circ=540^\\circ.[\/tex]<br \/>\nComo o pent\u00e1gono da figura \u00e9 regular, particularmente todos os seus \u00e2ngulos internos t\u00eam a mesma medida, portanto:<br \/>\n[tex]\\qquad \\qquad  \\textcolor{#339900}{\\beta=\\dfrac{540^\\circ}{5}}[\/tex]<\/li>\n<p>[tex]\\qquad \\qquad  \\textcolor{#339900}{\\boxed{\\beta=108^\\circ}}.[\/tex]<\/li>\n<li>O pol\u00edgono colorido de lil\u00e1s \u00e9 um oct\u00f3gono.<br \/>\nPela f\u00f3rmula citada no <span class=\"c3\"><strong>Lembrete<\/strong><\/span>, a soma dos \u00e2ngulos internos de um oct\u00f3gono \u00e9<br \/>\n[tex]\\qquad \\qquad S_8=\\left(8-2\\right)\\cdot 180^\\circ=1080^\\circ.[\/tex]<br \/>\nComo o oct\u00f3gono da figura \u00e9 regular, tamb\u00e9m todos os seus \u00e2ngulos internos t\u00eam a mesma medida, logo:<br \/>\n[tex]\\qquad \\qquad  \\textcolor{#CC66FF}{\\theta=\\dfrac{1080^\\circ}{8}}[\/tex]<\/li>\n<p>[tex]\\qquad \\qquad  \\textcolor{#CC66FF}{\\boxed{\\theta=135^\\circ}}.[\/tex]<\/li>\n<\/ul>\n<p>A partir dessas duas medidas, podemos concluir que:<br \/>\n[tex]\\qquad \\qquad \\textcolor{red}{\\alpha}=360^\\circ-\\textcolor{#339900}{\\beta}-\\textcolor{#CC66FF}{\\theta}[\/tex]<br \/>\n[tex]\\qquad \\qquad \\textcolor{red}{\\alpha}=360^\\circ-\\textcolor{#339900}{108^\\circ}-\\textcolor{#CC66FF}{135^\\circ}[\/tex]<br \/>\n[tex]\\qquad \\qquad \\textcolor{red}{\\boxed{\\alpha=117^\\circ}}[\/tex].<\/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=\"g3\">\nParticipou da discuss\u00e3o o Clube <strong>OCTETO MATEM\u00c1TICO<\/strong>.<\/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\/10\/PAE_Dois-pol\u00edgonos.pdf\" title=\" Arquivo PDF\" download=\"Clubes_ PAE_Dois pol\u00edgonos.pdf\"><span style=\"color: #FFFAFA; font-family: georgia; font-size: 16px;\"><strong>Arquivo <\/strong><\/span><\/a>\n <\/p>\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>&#9999; Link do problema para dispositivos da Apple. Problema (A partir do 8\u00ba ano do E. F. &#8211; N\u00edvel de dificuldade: F\u00e1cil) (ONEM &#8211; 2010) Os dois pol\u00edgonos mostrados na figura s\u00e3o regulares, t\u00eam um lado em comum e s\u00e3o coplanares. Determinar a medida angular [tex]\\textcolor{red}{\\alpha}[\/tex], em graus. Lembrete A soma dos \u00e2ngulos internos de &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-dois-poligonos\/\">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-137743","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/137743","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=137743"}],"version-history":[{"count":13,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/137743\/revisions"}],"predecessor-version":[{"id":220586,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/137743\/revisions\/220586"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=137743"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}