{"id":27971,"date":"2015-01-31T19:33:39","date_gmt":"2015-01-31T21:33:39","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=27971"},"modified":"2022-06-28T01:48:10","modified_gmt":"2022-06-28T04:48:10","slug":"um-pouco-sobre-poligonos-estrelas","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/um-pouco-sobre-poligonos-poligonos-uma-primeira-definicao-2\/um-pouco-sobre-poligonos-estrelas\/","title":{"rendered":". Um pouco sobre pol\u00edgonos &#8211; Estrelas 1"},"content":{"rendered":"<hr style=\"height:2px;background-color:#8B5A2B;  margin-top: 20px; margin-bottom: 8px;border: 0 none\"\/>\n<hr style=\"height:1px;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 20px;border: 0 none\"\/>\n<p class=\"nome cm\" style=\"margin:10px;line-height:50px; font-size:54px;\"><strong>Um pouco sobre pol\u00edgonos<\/strong><\/p>\n<hr style=\"height:1px;background-color:#8B5A2B;  margin-top: 20px; margin-bottom: 8px;border: 0 none\"\/>\n<hr style=\"height:2px;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 40px;border: 0 none\"\/>\n<div style=\"padding-right: 10px; padding-left:50%;\">\n<div class=\"ref\">\n\tOs <em>applets<\/em> disponibilizados nesta p\u00e1gina poder\u00e3o ser utilizados <em>on-line <\/em> e <em>off-line<\/em><br \/>\n\tPara utiliz\u00e1-los <em> off-line<\/em>, basta copiar os arquivos disponibilizados abaixo de cada <em>applet<\/em> e abri-los no GeoGebra do seu computador ou tablet.<br \/>\n\tSe voc\u00ea n\u00e3o tem o GeoGebra instalado,  visite <a href=\"https:\/\/www.geogebra.org\/download?lang=pt\" title=\"site do GeoGebra\" target=\"_blank\"> <span style=\"color: #8B5A2B; font-family: verdana, Arial, Helvetica, sans-serif; font-size: 12px;\"> <strong>este site<\/strong> <\/span><\/a> e procure pela vers\u00e3o adequada ao seu dispositivo.\n<\/div>\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"gr1 cbmd\">\n<p class=\"ti1 cm\"  style=\"line-height:25px;margin:10px\">Estrelas, uma apresenta\u00e7\u00e3o informal <\/p>\n<hr style=\"height:2px;background-color:#8B5A2B; border: 0 none\"\/>\n&nbsp;<\/p>\n<div style=\"text-align:center;\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:610px;\">\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=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:480px;background-color: #CDBA96;\" > Que figuras podemos obter quando dividimos uma circunfer\u00eancia em <em><strong>n<\/strong><\/em> partes iguais e ligamos, consecutivamente, os <em><strong>n<\/strong><\/em> pontos que determinam essa divis\u00e3o por segmentos de reta, saltando de <em><strong>k<\/strong><\/em> em <em><strong>k<\/strong><\/em> pontos?\n<\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<div class=\"gr1 cbm2\">\n<p class=\"ti2 cm\">I &#8211; <em>Applets<\/em> para ajudar no entendimento do problema proposto&#8230;<\/p>\n<hr style=\"height:2px;background-color:#8B5A2B; margin-top: 5px; margin-bottom: 5px; border: 0 none\"\/>\n<hr style=\"height:1px;background-color:#8B5A2B;  margin-top: 5px; margin-bottom: 5px;border: 0 none\"\/>\n<p style=\"margin:40px\">\n<div class=\"gr1 cbm5\">\n<p class=\"ti3 cm\">* Observando uma circunfer\u00eancia dividida em 7 partes iguais * <\/p>\n<hr style=\"height:2px;background-color:#8B5A2B; border: 0 none\"\/>\n&nbsp;<\/p>\n<div style=\"text-align:center;\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:550px;\">\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=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:400px;background-color: #CDBA96;\" > Disponibilizamos alguns <em>applets<\/em> e <em>gifs<\/em>, para voc\u00ea visualizar as figuras que obtemos a partir dos pontos que dividem uma circunfer\u00eancia em sete partes iguais.<\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<div class=\"gr1 cbm7\">\n<p class=\"ti3 cm\"><em>Applet<\/em> 1<\/p>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<ul>\n<li>Para utilizar o <em>applet on-line<\/em>, clique no bot\u00e3o <strong>Mostrar o Applet<\/strong>.<\/li>\n<li>Para copiar o <em>applet<\/em> e utiliz\u00e1-lo <em>off-line<\/em>, clique no bot\u00e3o <strong>Arquivo 1<\/strong>.<\/li>\n<\/ul>\n<div style=\"text-align:center;\">\n<div>\n<input  value=\"Mostrar o Applet\" type=\"button\" class=\"bo1 bbgm\"  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';}\"\/>    <\/p>\n<div  style=\"display: none;\">\n<div style=\"border: 1px solid #8B5A2B; padding: 5px;padding-left: 20px; padding-right: 20px; font-size: medium; font-family: verdana, Arial, Helvetica, sans-serif; color: #616161; line-height: 25px; text-align:justify; margin-left: auto; margin-right: auto;\">\n&nbsp;<br \/>\n<span style=\"font-size:14px\">Aguarde o arquivo carregar e clique sucessivamente em cada quadradinho, de cima para baixo.<br \/>\nPara reiniciar a constru\u00e7\u00e3o, clique nas setinhas que aparecer\u00e3o no canto superior direito do <em>applet<\/em>.<\/span><\/p>\n<div style=\"text-align:center;\">\n\tPara carregar o applet, clique <a href=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/PDbofLXX\/width\/626\/height\/697\/border\/ffffff\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/true\/ld\/false\/sri\/true\/at\/auto\" target=\"ap1\"><span style=\"color: #8B5A2B;\"><strong>AQUI<\/strong><\/span><\/a><br \/>\n<iframe  name=\"ap1\"  width=\"630\" height=\"700\" style=\"border: 5px solid #8B5A2B;\"> <\/iframe><br \/>\n<span style=\"font-size:14px\">OBMEP_srdg, criado com o <a href=\"http:\/\/www.geogebra.org\/\" target=\"_blank\" >GeoGebra<\/a><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\" float:right;  width: 100px;\"> <input  class=\"bo2  bbgm\"  style=\" width: 100px;\"  type=button  onClick=\"parent.location='http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/estrelados_7.ggb'\" value='Arquivo 1' title=\" Arquivo do GeoGebra\"\/><\/p>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:40px\">\n<\/div>\n<\/div>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto;\">\n<tr class=\"bf\">\n<td style=\"border: 1px solid #FFFFFF; vertical-align:top; WIDTH:5%\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/seta3.gif\" alt=\"seta3\" width=\"30\" height=\"13\" style=\"border: 0px;\" class=\"aligncenter size-full wp-image-26936\" \/><\/td>\n<td style=\"border: 1px solid #FFFFFF; vertical-align:top; text-align:justify\">\n<span style=\"color: #616161; font-family: verdana, Arial, Helvetica, sans-serif; font-size: medium; line-height: 25px;\"> Voc\u00ea percebeu que, embora seja poss\u00edvel saltarmos de 1 em 1; de 2 em 2; de 3 em 3; de 4 em 4; de 5 em 5 e de 6 em 6 pontos, obtemos apenas tr\u00eas figuras diferentes?<br \/>\nCom o <em> applet<\/em> abaixo, voc\u00ea pode conferir as tr\u00eas figuras.<\/span><\/td>\n<\/tr>\n<\/table>\n<\/div>\n<p style=\"margin:20px\">\n<div class=\"gr1 cbm7\">\n<p class=\"ti3 cm\"><em>Applet<\/em> 2<\/p>\n<hr style=\"height:1px;background-color:#8B5A2B;  border: 0 none\"\/>\n<ul>\n<li>Para utilizar o <em>applet on-line<\/em>, clique no bot\u00e3o <strong>Mostrar o Applet<\/strong>.<\/li>\n<li>Para copiar o <em>applet<\/em> e utiliz\u00e1-lo <em>off-line<\/em>, clique no bot\u00e3o <strong>Arquivo 2<\/strong>.<\/li>\n<\/ul>\n<p style=\"margin:20px\">\n<div style=\"text-align:center;\">\n<div>\n<input  value=\"Mostrar o Applet\" type=\"button\" class=\"bo1  bbgm\" 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';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<div style=\"border: 1px solid #8B5A2B; padding: 5px;padding-left: 20px; padding-right: 20px; font-size: medium; font-family: verdana, Arial, Helvetica, sans-serif; color: #616161; line-height: 25px; text-align:justify\">\n&nbsp;<br \/>\n<span style=\"font-size:14px\">Aguarde o arquivo carregar e selecione as op\u00e7\u00f5es desejadas.<\/span><\/p>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<p>\tPara carregar o applet, clique <a href=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/gYBFzM6w\/width\/493\/height\/577\/border\/ffffff\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/true\/ld\/false\/sri\/false\/at\/auto\" target=\"ap2\"><span style=\"color: #8B5A2B;\"><strong>AQUI<\/strong><\/span><\/a><br \/>\n<iframe   name=\"ap2\"  width=\"600\" height=\"580\" style=\"border: 5px solid #8B5A2B;\"> <\/iframe><br \/>\n<span style=\"font-size:14px\">OBMEP_srdg, criado com o <a href=\"http:\/\/www.geogebra.org\/\" target=\"_blank\" >GeoGebra<\/a><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\" float:right;  width: 100px;\"><input  class=\"bo2  bbgm\"  style=\" width: 100px;\"  type=button  onClick=\"parent.location='http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/estrelados7_todos1.ggb'\" value='Arquivo 2' title=\" Arquivo do GeoGebra\"\/><\/div>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:40px\">\n<\/div>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<p><strong>Para uma visualiza\u00e7\u00e3o din\u00e2mica das tr\u00eas figuras do <em>applet<\/em> 2,<br \/>\n\tutilize o <em>gif<\/em> animado abaixo.<\/strong><\/p>\n<p style=\"margin:20px\">\n<div>\n<input  value=\"Mostrar o gif\" type=\"button\" class=\"bo1  bbgm\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar o gif';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Mostrar o gif';}\"\/>    <\/p>\n<div  style=\"display: none;\">\n&nbsp;<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/estrelados7_todosanimados1.gif\" style=\"border: 0px;\" alt=\"estrelados7_todosanimados1\" width=\"500\" height=\"470\" class=\"aligncenter size-full wp-image-26987\" \/><br \/>\n<span style=\"font-size:14px\">OBMEP_srdg, criado com o <a href=\"http:\/\/www.geogebra.org\/\" target=\"_blank\" >GeoGebra<\/a><\/span>\n<\/div>\n<p style=\"margin:20px\">\n<\/div>\n<\/div>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:20px\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:530px;\">\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=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:400px;background-color: #CDBA96;\" >  Que tal repetir esse processo a partir de uma circunfer\u00eancia dividida em 5 partes iguais?<br \/>\nPara isso, utilize os <em>applets<\/em> abaixo&#8230;<\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<\/table>\n<div class=\"gr1 cbm5\">\n<p class=\"ti3 cm\">* Observando uma circunfer\u00eancia dividida em 5 partes iguais *<\/p>\n<hr style=\"height:2px;background-color:#8B5A2B; border: 0 none\"\/>\n<p style=\"margin:20px\">\n<div class=\"gr1 cbm7\">\n<p class=\"ti3 cm\"><em>Applet<\/em> 3<\/p>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<ul>\n<li>Para utilizar o <em>applet on-line<\/em>, clique no bot\u00e3o <strong>Mostrar o Applet<\/strong>.<\/li>\n<li><em>applet<\/em> e utiliz\u00e1-lo <em>off-line<\/em>, clique no bot\u00e3o <strong>Arquivo 3<\/strong>.<\/li>\n<\/ul>\n<p style=\"margin:20px\">\n<div style=\"text-align:center;\">\n<div>\n<input  value=\"Mostrar o Applet\" type=\"button\" class=\"bo1  bbgm\" 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';}\"\/>    <\/p>\n<div  style=\"display: none;\">\n<div style=\"border: 1px solid #8B5A2B; padding: 5px;padding-left: 20px; padding-right: 20px; font-size: medium; font-family: verdana, Arial, Helvetica, sans-serif; color: #616161; line-height: 25px; text-align:justify\">\n&nbsp;<br \/>\n<span style=\"font-size:14px\">Aguarde o arquivo carregar e clique sucessivamente em cada quadradinho, de cima para baixo.<br \/>\nPara reiniciar a constru\u00e7\u00e3o, clique nas setinhas que aparecer\u00e3o no canto superior direito do <em>applet<\/em>.<\/span><\/p>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<p>\tPara carregar o applet, clique <a href=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/oKZiMFgA\/width\/650\/height\/674\/border\/ffffff\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/true\/ld\/false\/sri\/true\/at\/auto\" target=\"ap3\"><span style=\"color: #8B5A2B;\"><strong>AQUI<\/strong><\/span><\/a><br \/>\n<iframe   name=\"ap3\"  width=\"660\" height=\"680\" style=\"border: 5px solid #8B5A2B;\"> <\/iframe><br \/>\n<span style=\"font-size:14px\">OBMEP_srdg, criado com o <a href=\"http:\/\/www.geogebra.org\/\" target=\"_blank\" >GeoGebra<\/a><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\" float:right;  width: 100px;\"><input  class=\"bo2  bbgm\"  style=\" width: 100px;\"  type=button  onClick=\"parent.location='http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/estrelados_5.ggb'\" value='Arquivo 3' title=\" Arquivo do GeoGebra\"\/><\/div>\n<p style=\"margin:40px\">\n<\/div>\n<\/div>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto;\">\n<tr class=\"bf\">\n<td style=\"border: 1px solid #FFFFFF; vertical-align:top; WIDTH:5%\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/seta3.gif\" alt=\"seta3\" width=\"30\" height=\"13\" style=\"border: 0px;\" class=\"aligncenter size-full wp-image-26936\" \/><\/td>\n<td style=\"border: 1px solid #FFFFFF; vertical-align:top; text-align:justify\"> <span style=\"color: #616161; font-family: verdana, Arial, Helvetica, sans-serif; font-size: medium; line-height: 25px;\">Com o <em>applet<\/em> anterior, \u00e9 poss\u00edvel observar que podemos saltar de 1 em 1; de 2 em 2; de 3 em 3 e de 4 em 4 pontos, mas obtemos apenas duas figuras diferentes.<br \/>\nCom o pr\u00f3ximo <em>applet<\/em>, voc\u00ea pode conferir as duas figuras&#8230;<\/span><\/td>\n<\/tr>\n<\/table>\n<\/div>\n<p style=\"margin:20px\">\n<div class=\"gr1 cbm7\">\n<p class=\"ti3 cm\"><em>Applet<\/em> 4<\/p>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<ul>\n<li>Para utilizar o <em>applet on-line<\/em>, clique no bot\u00e3o <strong>Mostrar o Applet<\/strong>.<\/li>\n<li>Para copiar o <em>applet<\/em> e utiliz\u00e1-lo <em>off-line<\/em>, clique no bot\u00e3o <strong>Arquivo 4<\/strong>.<\/li>\n<\/ul>\n<p style=\"margin:20px\">\n<div style=\"text-align:center;\">\n<div>\n<input  value=\"Mostrar o Applet\" type=\"button\" class=\"bo1  bbgm\" 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';}\"\/>    <\/p>\n<div  style=\"display: none;\">\n<div style=\"border: 1px solid #8B5A2B; padding: 5px;padding-left: 20px; padding-right: 20px; font-size: medium; font-family: verdana, Arial, Helvetica, sans-serif; color: #616161; line-height: 25px; text-align:justify\">\n&nbsp;<br \/>\n<span style=\"font-size:14px\">Aguarde o arquivo carregar e selecione as op\u00e7\u00f5es desejadas.<\/span><\/p>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<p>\tPara carregar o applet, clique <a href=\"https:\/www.geogebra.org\/material\/iframe\/id\/JxWXHwNd\/width\/576\/height\/504\/border\/ffffff\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/true\/ld\/false\/sri\/false\/at\/auto\" target=\"ap4\"><span style=\"color: #8B5A2B;\"><strong>AQUI<\/strong><\/span><\/a><br \/>\n<iframe  name=\"ap4\"  width=\"580\" height=\"505\" style=\"border: 5px solid #8B5A2B;\"> <\/iframe><br \/>\n<span style=\"font-size:14px\">OBMEP_srdg, criado com o <a href=\"http:\/\/www.geogebra.org\/\" target=\"_blank\" >GeoGebra<\/a><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\" float:right;  width: 100px;\"><input  class=\"bo2  bbgm\"  style=\" width: 100px;\"  type=button  onClick=\"parent.location='http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2018\/10\/estrelados5_todos.ggb'\" value='Arquivo 4' title=\" Arquivo do GeoGebra\"\/><\/div>\n<p style=\"margin:40px\">\n<\/div>\n<\/div>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<p><strong>Para uma visualiza\u00e7\u00e3o din\u00e2mica das duas figuras do <em>applet<\/em> 4,<br \/>\n\tutilize o <em>gif<\/em> animado abaixo.<\/strong><\/p>\n<p style=\"margin:20px\">\n<div>\n<input  value=\"Mostrar o gif\" type=\"button\" class=\"bo1  bbgm\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar o gif';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Mostrar o gif';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n&nbsp;<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/estrelados5_todos_animados.gif\" style=\"border: 0px;\"  alt=\"estrelados5_todos_animados\" width=\"384\" height=\"380\" class=\"aligncenter size-full wp-image-27179\" \/><br \/>\n<span style=\"font-size:14px\">OBMEP_srdg, criado com o <a href=\"http:\/\/www.geogebra.org\/\" target=\"_blank\" >GeoGebra<\/a><\/span>\n<\/div>\n<p style=\"margin:20px\">\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div style=\"text-align:center;\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:546px;\">\n<tr>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:390px;background-color: #CDBA96;\" > Que figuras bonitas!<br \/>\nMas ser\u00e1 que, dividindo uma circunfer\u00eancia em um n\u00famero qualquer de partes iguais e pulando do jeito que a gente quer, vamos sempre obter figuras como essas?<\/p>\n<p style=\"margin:20px\">\n<\/td>\n<td class=\"bf\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/02\/carinha6.jpg\" style=\"border: 0px;\" alt=\"carinha6\" width=\"96\" height=\"110\" class=\"alignleft size-full wp-image-3062\" \/><\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:556px;\">\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=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"background-color: #CDBA96; width:400px;\">  Essas figuras que chamaram a sua aten\u00e7\u00e3o s\u00e3o chamadas de <strong><em>estrelas<\/em><\/strong>. E pensando um pouquinho mais, talvez voc\u00ea mesmo consiga responder \u00e0 sua pergunta.<br \/>\nOs pr\u00f3ximos <em>applets<\/em> podem lhe ajudar com as respostas&#8230;<\/p>\n<p style=\"margin:40px\">\n<p style=\"margin:20px\">\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<div class=\"gr1 cbm5\">\n<p class=\"ti3 cm\">* Observando uma circunfer\u00eancia dividida em 8 partes iguais *<\/p>\n<hr style=\"height:2px;background-color:#8B5A2B; border: 0 none\"\/>\n<p style=\"margin:20px\">\n<div class=\"gr1 cbm7\">\n<p class=\"ti3 cm\"><em>Applet<\/em> 5<\/p>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<ul>\n<li>Para utilizar o <em>applet on-line<\/em>, clique no bot\u00e3o <strong>Mostrar o Applet<\/strong>.<\/li>\n<li>Para copiar o <em>applet<\/em> e utiliz\u00e1-lo <em>off-line<\/em>, clique no bot\u00e3o <strong>Arquivo 5<\/strong>.<\/li>\n<\/ul>\n<p style=\"margin:20px\">\n<div style=\"text-align:center;\">\n<div>\n<input  value=\"Mostrar o Applet\" type=\"button\" class=\"bo1  bbgm\" 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';}\"\/>    <\/p>\n<div   style=\"display: none;\">\n<div style=\"border: 1px solid #8B5A2B; padding: 5px;padding-left: 20px; padding-right: 20px; font-size: medium; font-family: verdana, Arial, Helvetica, sans-serif; color: #616161; line-height: 25px; text-align:justify\">\n&nbsp;<br \/>\n<span style=\"font-size:14px\">Aguarde o arquivo carregar e clique sucessivamente em cada quadradinho, de cima para baixo.<br \/>\nPara reiniciar a constru\u00e7\u00e3o, clique nas setinhas que aparecer\u00e3o no canto superior direito do <em>applet<\/em>.<\/span><\/p>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<p>\tPara carregar o applet, clique <a href=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/nJDq3W2o\/width\/633\/height\/571\/border\/ffffff\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/true\/ld\/false\/sri\/true\/at\/auto\" target=\"ap5\"><span style=\"color: #8B5A2B;\"><strong>AQUI<\/strong><\/span><\/a><br \/>\n<iframe   name=\"ap5\"  width=\"650\" height=\"575\" style=\"border: 5px solid #8B5A2B;\"> <\/iframe><br \/>\n<span style=\"font-size:14px\">OBMEP_srdg, criado com o <a href=\"http:\/\/www.geogebra.org\/\" target=\"_blank\" >GeoGebra<\/a><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\" float:right;  width: 100px;\"><input  class=\"bo2  bbgm\"  style=\" width: 100px;\"  type=button  onClick=\"parent.location='http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/estrelados_81.ggb'\" value='Arquivo 5' title=\" Arquivo do GeoGebra\"\/><\/div>\n<p style=\"margin:20px\">\n<p><span style=\"line-height: 5px;\"><\/span>\n<\/div>\n<p style=\"margin:40px\">\n<\/div>\n<p style=\"margin:20px\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto;\">\n<tr class=\"bf\">\n<td style=\"border: 1px solid #FFFFFF; vertical-align:top; WIDTH:5%\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/seta3.gif\" alt=\"seta3\" width=\"30\" height=\"13\" style=\"border: 0px;\" class=\"aligncenter size-full wp-image-26936\" \/><\/td>\n<td style=\"border: 1px solid #FFFFFF; vertical-align:top; text-align:justify\"><span style=\"color: #616161; font-family: verdana, Arial, Helvetica, sans-serif; font-size: medium; line-height: 25px;\">Neste caso, ao pularmos de 4 em 4 pontos n\u00e3o aparecem figuras fechadas, pois obtemos v\u00e1rios segmentos de reta que concorrem em um mesmo ponto que n\u00e3o \u00e9 extremidade de nenhum dos segmentos. Mas, aparentemente, ao saltarmos de 1 em 1; de 2 em 2; de 3 em 3; de 5 em 5, de 6 em 6 e de 7 em 7 pontos obtemos figuras <em>fechadas <\/em> que se repetem, como nas situa\u00e7\u00f5es anteriores.<br \/>\nNo pr\u00f3ximo <em>applet<\/em>, registramos essas aparentes figuras fechadas.<\/span><\/td>\n<\/tr>\n<\/table>\n<p style=\"margin:20px\">\n<div class=\"gr1 cbm7\">\n<p class=\"ti3 cm\"><em>Applet<\/em> 6<\/p>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<ul>\n<li>Para utilizar o <em>applet on-line<\/em>, clique no bot\u00e3o <strong>Mostrar o Applet<\/strong>.<\/li>\n<li>Para copiar o <em>applet<\/em> e utiliz\u00e1-lo <em>off-line<\/em>, clique no bot\u00e3o <strong>Arquivo 6<\/strong>.<\/li>\n<\/ul>\n<p style=\"margin:20px\">\n<div style=\"text-align:center;\">\n<div>\n<input  value=\"Mostrar o Applet\" type=\"button\" class=\"bo1  bbgm\" 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';}\"\/>    <\/p>\n<div style=\"display: none;\">\n<div style=\"border: 1px solid #8B5A2B; padding: 5px;padding-left: 20px; padding-right: 20px; font-size: 16px; font-family: verdana, Arial, Helvetica, sans-serif; color: #616161; line-height: 25px;text-align:justify\">\n&nbsp;<br \/>\n<span style=\"font-size:14px\">Aguarde o arquivo carregar e selecione as op\u00e7\u00f5es desejadas.<br \/>\n\tEm particular, observe a figura resultante de pularmos de 2 em 2 pontos!<br \/>\n\t<\/span><\/p>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<p>\tPara carregar o applet, clique <a href=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/CkfPVhUa\/width\/535\/height\/534\/border\/ffffff\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/true\/ld\/false\/sri\/false\/at\/auto\" target=\"ap6\"><span style=\"color: #8B5A2B;\"><strong>AQUI<\/strong><\/span><\/a><br \/>\n<iframe   name=\"ap6\" width=\"550\" height=\"535\" style=\"border: 5px solid #8B5A2B;\"> <\/iframe><br \/>\n<span style=\"font-size:14px\">OBMEP_srdg, criado com o <a href=\"http:\/\/www.geogebra.org\/\" target=\"_blank\" >GeoGebra<\/a><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\" float:right;  width: 100px;\"><input  class=\"bo2  bbgm\"  style=\" width: 100px;\"  type=button  onClick=\"parent.location='http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/estrelados_8_todos.ggb'\" value='Arquivo 6' title=\" Arquivo do GeoGebra\"\/><\/div>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:40px\">\n<\/div>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<p><strong>Para uma visualiza\u00e7\u00e3o din\u00e2mica das tr\u00eas figuras do <em>applet<\/em> 6, utilize o <em>gif<\/em> animado abaixo.<br \/>\n\tObserve mais uma vez a figura resultante ao pularmos de 2 em 2 pontas<\/strong><\/p>\n<p style=\"margin:20px\">\n<div>\n<input  value=\"Mostrar o gif\" type=\"button\" class=\"bo1  bbgm\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar o gif';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Mostrar o gif';}\"\/>    <\/p>\n<div  style=\"display: none;style=\"text-align:center;\"\">\n&nbsp;<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/estrelados_8_animados.gif\" style=\"border: 0px;\" alt=\"estrelados_8_animados\" width=\"403\" height=\"387\" class=\"aligncenter size-full wp-image-27264\" \/><\/p>\n<p style=\"text-align:center\"><span style=\"font-size:14px\">OBMEP_srdg, criado com o <a href=\"http:\/\/www.geogebra.org\/\" target=\"_blank\" >GeoGebra<\/a><\/span><\/p>\n<\/div>\n<p style=\"margin:20px\">\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px\">\n<div style=\"text-align:center;\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:418px;\">\n<tr>\n<td class=\"bf\">\n<p class=\"fala\" style=\"background-color: #CDBA96; width:300px;\"> Pulos que n\u00e3o definem figuras fechadas?<br \/>\nEstrelas falsas?<br \/>\nComplicou&#8230; <\/p>\n<p style=\"margin:20px\">\n<\/td>\n<td class=\"bf\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2014\/04\/carinha19.jpg\" style=\"border: 0px;\" alt=\"carinha19\" width=\"90\" height=\"91\" class=\"alignleft size-full wp-image-17463\" \/><\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:555px;\">\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=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:390px;background-color: #CDBA96;\" >  Calma, n\u00e3o \u00e9 t\u00e3o complicado quanto parece&#8230;<br \/>\nVamos observar o que acontece quando dividimos uma circunfer\u00eancia em 9 partes iguais (arcos congruentes), para depois fazermos algumas observa\u00e7\u00f5es. Neste caso tamb\u00e9m teremos estrelas falsas&#8230;<\/p>\n<p style=\"margin:20px\">\n<p><\/td>\n<\/tr>\n<\/table>\n<\/div>\n<div class=\"gr1 cbm5\">\n<p class=\"ti3 cm\">* Observando uma circunfer\u00eancia dividida em 9 partes iguais *<\/p>\n<hr style=\"height:2px;background-color:#8B5A2B; border: 0 none\"\/>\n<p style=\"margin:20px\">\n<div class=\"gr1 cbm7\">\n<p class=\"ti3 cm\"><em>Applet<\/em> 7<\/p>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<ul>\n<li>Para utilizar o <em>applet on-line<\/em>, clique no bot\u00e3o <strong>Mostrar o Applet<\/strong>.<\/li>\n<li>Para copiar o <em>applet<\/em> e utiliz\u00e1-lo <em>off-line<\/em>, clique no bot\u00e3o <strong>Arquivo 7<\/strong>.<\/li>\n<\/ul>\n<p style=\"margin:20px\">\n<div style=\"text-align:center;\">\n<div>\n<input value=\"Mostrar o Applet\" type=\"button\" class=\"bo1  bbgm\" 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';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<div style=\"border: 1px solid #8B5A2B; padding: 5px;padding-left: 20px; padding-right: 20px; font-size: medium; font-family: verdana, Arial, Helvetica, sans-serif; color: #616161; line-height: 25px; text-align:justify\">\n&nbsp;<br \/>\n<span style=\"font-size:14px\">Aguarde o arquivo carregar e clique sucessivamente em cada quadradinho, de modo a construir a circunfer\u00eancia, dividi-la em partes iguais e saltar em ordem crescente de pulos.<br \/>\nPara reiniciar a constru\u00e7\u00e3o, clique nas setinhas que aparecer\u00e3o no canto superior direito do <em>applet<\/em>.<\/span><\/p>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<p>\tPara carregar o applet, clique <a href=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/pYJYe07M\/width\/649\/height\/575\/border\/ffffff\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/true\/ld\/false\/sri\/true\/at\/auto\" target=\"ap7\"><span style=\"color: #8B5A2B;\"><strong>AQUI<\/strong><\/span><\/a><br \/>\n<iframe  name=\"ap7\"  width=\"650\" height=\"575\" style=\"border: 5px solid #8B5A2B;\"> <\/iframe><br \/>\n<span style=\"font-size:14px\">OBMEP_srdg, criado com o <a href=\"http:\/\/www.geogebra.org\/\" target=\"_blank\" >GeoGebra<\/a><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\" float:right;  width: 100px;\"><input  class=\"bo2  bbgm\"  style=\" width: 100px;\"  type=button  onClick=\"parent.location='http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/estrelados_91.ggb'\" value='Arquivo 7' title=\" Arquivo do GeoGebra\"\/><\/div>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:40px\">\n<\/div>\n<p style=\"margin:20px\">\n<table  class=\"tab1\">\n<tr class=\"bf\">\n<td style=\"border: 1px solid #FFFFFF; vertical-align:top; WIDTH:5%\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/seta3.gif\" alt=\"seta3\" width=\"30\" height=\"13\" style=\"border: 0px;\" class=\"aligncenter size-full wp-image-26936\" \/><\/td>\n<td style=\"border: 1px solid #FFFFFF; vertical-align:top; text-align:justify\"> <span style=\"color: #616161; font-family: verdana, Arial, Helvetica, sans-serif; font-size: medium; line-height: 25px;\">Aqui, podemos saltar de 1 em 1; de 2 em 2; de 3 em 3; de 4 em 4; de 5 em 5, de 6 em 6, de 7 em 7 e de 8 em 8 pontos: em todos os casos obteremos figuras aparentemente fechadas que podem se repetir.<br \/>\nNo pr\u00f3ximo <em>applet<\/em>, registramos as figuras.<\/span><\/td>\n<\/tr>\n<\/table>\n<p style=\"margin:20px\">\n<div class=\"gr1 cbm7\">\n<p class=\"ti3 cm\"><em>Applet<\/em> 8<\/p>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<ul>\n<li>Para utilizar o <em>applet on-line<\/em>, clique no bot\u00e3o <strong>Mostrar o Applet<\/strong>.<\/li>\n<li>Para copiar o <em>applet<\/em> e utiliz\u00e1-lo <em>off-line<\/em>, clique no bot\u00e3o <strong>Arquivo 8<\/strong>.<\/li>\n<\/ul>\n<p style=\"margin:20px\">\n<div style=\"text-align:center;\">\n<input value=\"Mostrar o Applet\" type=\"button\" class=\"bo1  bbgm\" 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';}\"\/>    <\/p>\n<div  style=\"display: none;\">\n<div style=\"border: 1px solid #8B5A2B; padding: 5px;padding-left: 20px; padding-right: 20px; font-size: medium; font-family: verdana, Arial, Helvetica, sans-serif; color: #616161; line-height: 25px;text-align:justify\">\n&nbsp;<br \/>\n<span style=\"font-size:14px\">Aguarde o arquivo carregar e selecione as op\u00e7\u00f5es desejadas.<br \/>\n\tObserve cuidadosamente a figura resultante de pularmos de 3 em 3 pontos (\u00faltima figura).<br \/>\n\t<\/span><\/p>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<p>\tPara carregar o applet, clique <a href=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/qDgdyuGb\/width\/524\/height\/601\/border\/ffffff\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/true\/ld\/false\/sri\/false\/at\/auto\" target=\"ap8\"><span style=\"color: #8B5A2B;\"><strong>AQUI<\/strong><\/span><\/a><br \/>\n<iframe name=\"ap8\"  width=\"530\" height=\"601\" style=\"border: 5px solid #8B5A2B;\"> <\/iframe><br \/>\n<span style=\"font-size:14px\">OBMEP_srdg, criado com o <a href=\"http:\/\/www.geogebra.org\/\" target=\"_blank\" >GeoGebra<\/a><\/span><\/div>\n<\/div>\n<\/div>\n<div style=\" float:right;  width: 100px;\"><input  class=\"bo2  bbgm\"  style=\" width: 100px;\"  type=button  onClick=\"parent.location='http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/estrelados_9_todos.ggb'\" value='Arquivo 8' title=\" Arquivo do GeoGebra\"\/><\/div>\n<p style=\"margin:40px\">\n<p><span style=\"line-height: 5px;\"><\/span>\n<\/div>\n<\/div>\n<p style=\"margin:20px\">\n<\/p><\/div>\n<p style=\"margin:20px\">\n<div style=\"text-align:center;\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:549px;\">\n<tr>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:390px;background-color: #CDBA96;\" >  Esse \u00faltimo caso me mostrou que o aparecimento das chamadas estrelas falsas (ainda n\u00e3o entendi bem o que \u00e9 isso&#8230;) n\u00e3o decorre de dividirmos a circunfer\u00eancia em um n\u00famero <strong>par <\/strong>de arcos iguais.<br \/>\nApareceram estrelas falsas quando dividimos a circunfer\u00eancia em 8 partes iguais, mas tamb\u00e9m quando dividimos a circunfer\u00eancia em 9 partes.<\/p>\n<p style=\"margin:20px\">\n<\/td>\n<td class=\"bf\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/02\/carinha3.jpg\" style=\"border: 0px;\" alt=\"carinha3\" width=\"90\" height=\"88\" class=\"alignleft size-full wp-image-3059\" \/><\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:555px;\">\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=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:390px;background-color: #CDBA96;\" >  Isso mesmo!<br \/>\nVamos, ent\u00e3o, registrar algumas observa\u00e7\u00f5es sobre os quatro casos apresentados,<br \/>\n<span style=\"padding-left: 40px;\"> <\/span><strong>circunfer\u00eancias divididas em 5, 7, 8 e 9 partes iguais<\/strong>,<br \/>\npara tentarmos responder \u00e0 pergunta inicial proposta.<\/p>\n<p><\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/div>\n<p style=\"margin:30px\">\n<hr style=\"height:1px; width:60%;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 8px;border: 0 none\"\/>\n<hr style=\"height:2px; width:80%;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 8px;border: 0 none\"\/>\n<hr style=\"height:3px; width:90%;background-color:#8B5A2B;   margin-top: 8px; margin-bottom: 8px;border: 0 none\"\/>\n<hr style=\"height:3px; width:90%;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 8px; border: 0 none\"\/>\n<hr style=\"height:2px; width:80%;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 8px;border: 0 none\"\/>\n<hr style=\"height:1px; width:60%;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 8px;border: 0 none\"\/>\n<p style=\"margin:30px\">\n<div class=\"gr1 cbm2\">\n<p class=\"ti2 cm\">II &#8211; Respondendo ao problema proposto<\/p>\n<hr style=\"height:2px;background-color:#8B5A2B;   margin-top: 5px; margin-bottom: 5px;border: 0 none\"\/>\n<hr style=\"height:1px;background-color:#8B5A2B;   margin-top: 5px; margin-bottom: 5px;border: 0 none\"\/>\n<div style=\"text-align:center;\">\n<p style=\"margin:20px\">\n<p style=\"text-align: justify;padding: 10px; border-style: solid; border-color :#8B5A2B; border-width:2px; width:80%\"><span style=\"color: #1C1C1C; font-family: verdana, Arial, Helvetica, sans-serif; font-size: 14px;\"> <strong>Problema<\/strong><br \/>\nSe [tex]A_1,\\,  A_2,\\,  \\cdots , A_n [\/tex] s\u00e3o pontos que dividem uma dada circunfer\u00eancia em  [tex]n[\/tex]  partes iguais, que figuras podemos obter se ligarmos por segmentos de reta esses  [tex]n[\/tex]  pontos, consecutivamente, de [tex]k[\/tex] em  [tex]k[\/tex]  pontos? <\/span><\/p>\n<\/div>\n<p>Na nossa discuss\u00e3o,  [tex]n[\/tex]  representar\u00e1 o n\u00famero de partes iguais (arcos congruentes) em que uma dada circunfer\u00eancia foi dividida, enquanto  [tex]k[\/tex]  corresponder\u00e1 ao n\u00famero de saltos feitos para tra\u00e7armos segmentos de reta ligando pontos dessa divis\u00e3o.<br \/>\nVamos, inicialmente, reunir na tabela abaixo algumas das informa\u00e7\u00f5es que podemos obter com os <em>applets<\/em> que foram aqui disponibilizados. <\/p>\n<p style=\"margin:20px\">\n<div style=\"text-align:center;\">\n<table style=\"border:2px solid;  width:500px; margin-left: auto; margin-right: auto;\">\n<tr>\n<td   style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">&nbsp;&nbsp;&nbsp;&nbsp;<strong> [tex]n[\/tex] <\/strong>&nbsp;&nbsp;&nbsp;&nbsp;<\/td>\n<td   style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">&nbsp;&nbsp;&nbsp;&nbsp;<strong> [tex]k[\/tex] &nbsp;&nbsp;&nbsp;&nbsp;<\/strong><\/td>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\"><strong>figura obtida<\/strong><\/td>\n<\/tr>\n<tr>\n<td  rowspan=4  style=\" border: 2px solid;  font-size: medium;\">5<\/td>\n<td   style=\" border: 2px solid;  font-size: medium; \">1<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; \">pent\u00e1gono regular<\/td>\n<\/tr>\n<tr>\n<td  style=\" border: 2px solid;  font-size: medium; \">2<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; \">estrela de cinco pontas (pentagrama)<\/td>\n<\/tr>\n<tr>\n<td style=\" border: 2px solid;  font-size: medium; \">3<\/td>\n<td style=\" border: 2px solid;  font-size: medium; \">estrela de cinco pontas (pentagrama)<\/td>\n<\/tr>\n<tr>\n<td  style=\" border: 2px solid;  font-size: medium; \">4<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; \">pent\u00e1gono regular<\/td>\n<\/tr>\n<tr>\n<td rowspan=6   style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">7<\/td>\n<td style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">1<\/td>\n<td style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">hept\u00e1gono regular<\/td>\n<\/tr>\n<tr>\n<td style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">2<\/td>\n<td style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">estrela de sete pontas 1<\/td>\n<\/tr>\n<tr>\n<td style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">3<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">estrela de sete pontas 2<\/td>\n<\/tr>\n<tr>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">4<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">estrela de sete pontas 2<\/td>\n<\/tr>\n<tr>\n<td   style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">5<\/td>\n<td   style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">estrela de sete pontas 1<\/td>\n<\/tr>\n<tr>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">6<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">hept\u00e1gono regular<\/td>\n<\/tr>\n<tr>\n<td  rowspan=7 style=\" border: 2px solid;  font-size: medium;\">8<\/td>\n<td   style=\" border: 2px solid;  font-size: medium; \">1<\/td>\n<td   style=\" border: 2px solid;  font-size: medium; \">oct\u00f3gono regular<\/td>\n<\/tr>\n<tr>\n<td   style=\" border: 2px solid;  font-size: medium; \">2<\/td>\n<td   style=\" border: 2px solid;  font-size: medium; \">estrela de oito pontas falsa<\/td>\n<\/tr>\n<tr>\n<td  style=\" border: 2px solid;  font-size: medium; \">3<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; \">estrela de oito pontas 1 <\/td>\n<\/tr>\n<tr>\n<td style=\" border: 2px solid;  font-size: medium; \">4<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; \">segmentos de reta concorrentes <\/td>\n<\/tr>\n<tr>\n<td   style=\" border: 2px solid;  font-size: medium; \">5<\/td>\n<td   style=\" border: 2px solid;  font-size: medium; \">estrela de oito pontas 1 <\/td>\n<\/tr>\n<tr>\n<td   style=\" border: 2px solid;  font-size: medium; \">6<\/td>\n<td   style=\" border: 2px solid;  font-size: medium; \">estrela de oito pontas falsa<\/td>\n<\/tr>\n<tr>\n<td style=\" border: 2px solid;  font-size: medium; \">7<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; \">oct\u00f3gono regular<\/td>\n<\/tr>\n<tr>\n<td rowspan=8 style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">9<\/td>\n<td style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">1<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">hept\u00e1gono regular<\/td>\n<\/tr>\n<tr>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">2<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">estrela de nove pontas 1<\/td>\n<\/tr>\n<tr>\n<td style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">3<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">estrela de nove pontas falsa<\/td>\n<\/tr>\n<tr>\n<td style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">4<\/td>\n<td style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">estrela de nove pontas 2<\/td>\n<\/tr>\n<tr>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">5<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">estrela de nove pontas 2<\/td>\n<\/tr>\n<tr>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">6<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">estrela de nove pontas falsa<\/td>\n<\/tr>\n<tr>\n<td style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">7<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">estrela de nove pontas 1<\/td>\n<\/tr>\n<tr>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">8<\/td>\n<td  style=\" border: 2px solid;  font-size: medium; background-color:#CFCFCF;\">hept\u00e1gono regular<\/td>\n<\/tr>\n<\/table>\n<p style=\"margin:20px\">\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:520px;\">\n<tr class=\"bf\">\n<td   class=\"bf\">\n<p class=\"fala\" style=\"background-color: #CDBA96; width:380px;\">  Como eu n\u00e3o entendi o que s\u00e3o estrelas falsas, eu responderia \u00e0 pergunta do problema dizendo que, quando \u00e9 formada uma figura fechada, essa figura \u00e9 um pol\u00edgono regular (quando damos os maiores ou os menores pulos) ou uma estrela. Mas podem aparecer segmentos de reta&#8230; <\/p>\n<p style=\"margin:20px\">\n<\/td>\n<td class=\"bf\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2014\/04\/carinha20.jpg\" alt=\"carinha20\" width=\"85\" height=\"82\" class=\"alignleft size-full wp-image-17461\" \/><\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:550px;\">\n<tr class=\"bf\">\n<td style=\"border: 1px solid #ffffff\"><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=\"95\" height=\"115\" \/><\/td>\n<td  style=\"border: 1px solid  #ffffff\">\n<p class=\"fala\" style=\"background-color: #CDBA96; width:400px;\"> Muito boa a sua observa\u00e7\u00e3o com rela\u00e7\u00e3o ao aparecimento de pol\u00edgonos regulares!<br \/>\nCom rela\u00e7\u00e3o \u00e0s estrelas falsas, o problema \u00e9 o que chamamos de <em> figura fechada<\/em>, na Matem\u00e1tica&#8230;<br \/>\nObserve atentamente as duas situa\u00e7\u00f5es apresentadas nas quais apareceram estrelas falsas: oito e nove pontas.<\/p>\n<p style=\"margin:20px\">\n<\/td>\n<\/tr>\n<\/table>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/8.gif\" alt=\"8\" width=\"335\" height=\"233\" class=\"aligncenter size-full wp-image-27691\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/9.gif\" alt=\"9\" width=\"541\" height=\"254\" class=\"aligncenter size-full wp-image-27692\" \/><\/p>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<p style=\"margin:20px\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:550px;\">\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=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"background-color: #CDBA96; width:400px;\"> Em cada &#8220;estrela&#8221;, caminhe sobre seus lados, a partir de uma de suas pontas.<br \/>\nO que aconteceu?<\/p>\n<p style=\"margin:20px\">\n<\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:540px;\">\n<tr>\n<td class=\"bf\">\n<p class=\"fala\" style=\"background-color: #CDBA96; width:390px;\">  Nas chamadas estrelas, partimos de uma de suas pontas e, sem tirar o l\u00e1pis do papel onde a figura est\u00e1 desenhada, passamos por todos os seus lados e chegamos na ponta da qual partimos. Nas estrelas falsas, para voltarmos para a ponta inicial passando por todos os seus lados, temos que tirar o l\u00e1pis do papel pelo menos uma vez&#8230;<br \/>\n&#201; isso? <\/p>\n<p style=\"margin:20px\">\n<\/td>\n<td class=\"bf\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/02\/carinha3.jpg\" style=\"border: 0px;\" alt=\"carinha3\" width=\"87\" height=\"85\" class=\"alignleft size-full wp-image-3059\" \/><\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:550px;\">\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=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"background-color: #CDBA96; width:400px;\"> Muito bem, voc\u00ea est\u00e1 corret\u00edssimo!<br \/>\nNa verdade, as estrelas falsas n\u00e3o s\u00e3o o que classificamos na matem\u00e1tica como figuras fechadas; elas s\u00e3o sobreposi\u00e7\u00f5es de pol\u00edgonos regulares.<br \/>\nObserve.<\/p>\n<p style=\"margin:20px\">\n<\/td>\n<\/tr>\n<\/table>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/8_1.gif\" alt=\"8_1\" width=\"257\" height=\"215\" class=\"aligncenter size-full wp-image-27701\" \/><span style=\"color: #8B5A2B; font-family: Georgia, Times, 'Times New Roman', serif; font-size: medium;\"> <strong>A estrela falsa de oito pontas \u00e9 a sobreposi\u00e7\u00e3o de dois quadrados,<br \/>\nlogo n\u00e3o consideramos <em>uma <\/em>figura fechada, propriamente dita.<\/strong><\/span><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/9_1.gif\" alt=\"9_1\" width=\"229\" height=\"242\" class=\"aligncenter size-full wp-image-27708\" \/><span style=\"color: #8B5A2B; font-family: Georgia, Times, 'Times New Roman', serif; font-size: medium;\"> <strong>A estrela falsa de nove pontas \u00e9 a sobreposi\u00e7\u00e3o de tr\u00eas tri\u00e2ngulos equil\u00e1teros,<br \/>\nportanto tamb\u00e9m n\u00e3o \u00e9 considerada <em>uma <\/em>figura fechada.<\/strong><\/span><\/p>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<p style=\"margin:20px\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:550px;\">\n<tr class=\"bf\">\n<td style=\"border: 1px solid  #ffffff\">\n<p class=\"fala\" style=\"background-color: #CDBA96; width:400px;\">  Agora sim&#8230;<br \/>\nO que est\u00e1 sendo chamado de estrela, digamos verdadeira, \u00e9 aquela na qual partimos de uma de suas pontas e, percorrendo continuamente um \u00fanico caminho,<strong> passamos por todas as outras pontas  e voltamos para a ponta da qual partimos<\/strong>.<\/p>\n<p style=\"margin:20px\">\n<\/td>\n<td style=\"border: 1px solid #ffffff\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/02\/carinha6.jpg\" alt=\"carinha6\" width=\"90\" height=\"86\" class=\"alignleft size-full wp-image-3062\" \/><\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:555px;\">\n<tr class=\"bf\">\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=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:400px;background-color: #CDBA96;\" > Correto, parab\u00e9ns!<br \/>\nVamos ver mais um exemplo de estrela falsa: ela aparece quando dividimos uma circunfer\u00eancia em 6 partes iguais e tem, portanto, 6 pontas.<\/p>\n<p style=\"margin:20px\">\n<\/td>\n<\/tr>\n<\/table>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/01\/6.gif\" alt=\"6\" width=\"457\" height=\"238\" class=\"aligncenter size-full wp-image-27766\" \/><span style=\"color: #8B5A2B; font-family: Georgia, Times, 'Times New Roman', serif; font-size: medium;\"> <strong> Aqui temos a sobreposi\u00e7\u00e3o de dois tri\u00e2ngulos equil\u00e1teros.<br \/>\nPortanto, temos dois caminhos e, consequentemente, uma figura n\u00e3o fechada.<\/strong><\/span><\/p>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<p style=\"margin:20px\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:550px;\">\n<tr>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:400px;background-color: #CDBA96;\" > Bom, agora eu consigo responder \u00e0 pergunta proposta:<br \/>\nQuando unimos por segmentos de retas os pontos que dividem uma circunfer\u00eancia em  [tex]n[\/tex]  partes iguais, pulando de  [tex]k[\/tex]  em  [tex]k[\/tex]  pontos, podemos obter segmentos de retas concorrentes, pol\u00edgonos regulares, estrelas e falsas estrelas.<\/p>\n<p style=\"margin:20px\">\n<\/td>\n<td class=\"bf\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/02\/carinha1.jpg\" alt=\"carinha1\" width=\"100\" height=\"86\" class=\"alignleft size-full wp-image-3063\" \/><\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:477px;\">\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=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"background-color: #CDBA96; width:350px;\"> Vamos, ent\u00e3o, registrar, mais uma vez, a pergunta do problema, mas agora acompanhada de uma resposta.<\/p>\n<p style=\"margin:20px\">\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<hr style=\"height:1px;background-color:#8B5A2B; border: 0 none\"\/>\n<p style=\"margin:20px\">\n<div style=\"border: 2px dotted #8B5A2B; padding: 5px;padding-left: 35px; padding-right: 35px; font-size: 14px; font-family: verdana, Arial, Helvetica, sans-serif; color: #616161; line-height: 25px;text-align:justify\">\n<strong>Problema:<\/strong><\/p>\n<p style=\"margin-left:25px; margin-right:25px;\">\nSe [tex]A_1,\\,  A_2,\\,  \\cdots , A_n [\/tex] s\u00e3o pontos que dividem uma dada circunfer\u00eancia em  [tex]n[\/tex]  partes iguais, que figuras podemos obter se ligarmos por segmentos de reta esses  [tex]n[\/tex]  pontos, consecutivamente, de [tex]k[\/tex]  em  [tex]k[\/tex]  pontos?<\/p>\n<p><strong>Resposta:<\/strong><\/p>\n<p style=\"margin-left:25px; margin-right:25px;\">\nO problema s\u00f3 tem sentido se  [tex]n[\/tex]  for um n\u00famero natural tal que [tex] n &ge; 2[\/tex]>.<\/p>\n<ul>\n<li>Se [tex] n= 2[\/tex], a figura obtida \u00e9, necessariamente, um segmento de reta.<\/li>\n<li>Se [tex] n= 3[\/tex] a figura obtida \u00e9, necessariamente, um tri\u00e2ngulo equil\u00e1tero.<\/li>\n<li>Se [tex] n &gt; 3[\/tex], podem ser obtidas as seguintes figuras:<\/li>\n<ul>\n<li>pol\u00edgonos regulares;<\/li>\n<li>segmentos de reta concorrentes;<\/li>\n<li>estrelas;<\/li>\n<li>estrelas falsas (sobreposi\u00e7\u00e3o de pol\u00edgonos).<\/li>\n<\/ul>\n<\/ul>\n<\/div>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:30px\">\n<hr style=\"height:1px; width:60%;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 8px;border: 0 none;\"\/>\n<hr style=\"height:2px; width:80%;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 8px;border: 0 none\"\/>\n<hr style=\"height:3px; width:90%;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 8px;border: 0 none\"\/>\n<hr style=\"height:3px; width:90%;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 8px;border: 0 none\"\/>\n<hr style=\"height:2px; width:80%;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 8px;border: 0 none;\"\/>\n<hr style=\"height:1px; width:60%;background-color:#8B5A2B;  margin-top: 8px; margin-bottom: 8px;border: 0 none;\"\/>\n<p style=\"margin:30px\">\n<div class=\"gr1 cbm2\">\n<p class=\"ti2 cm\">III &#8211; Explorando o problema proposto <\/p>\n<hr style=\"height:2px;background-color:#8B5A2B;  margin-top: 5px; margin-bottom: 5px; border: 0 none;\"\/>\n<hr style=\"height:1px;background-color:#8B5A2B;   margin-top: 5px; margin-bottom: 5px;border: 0 none;\"\/>\n<p style=\"margin:20px\">\n<div style=\"text-align:center;\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:553px;\">\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=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"background-color: #CDBA96; width:400px;\"> Se formos dividir uma circunfer\u00eancia em <strong><em>n<\/em><\/strong> partes iguais, podemos prever de quantos em quantos pontos devemos pular para obtermos estrelas (n\u00e3o falsas) com <strong><em>n<\/em><\/strong> pontas, mesmo antes de fazermos a divis\u00e3o?<\/p>\n<p style=\"margin:20px\">\n<\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:389px;\">\n<tr>\n<td class=\"bf\">\n<p class=\"fala\" style=\"background-color: #CDBA96; width:230px;\">  Nossa!!!!!!<br \/>\nSer\u00e1 que isso \u00e9 poss\u00edvel ?!?!<\/p>\n<p style=\"margin:20px\">\n<\/td>\n<td class=\"bf\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/02\/Desafio2.png\" alt=\"Desafio2\" width=\"130\" height=\"110\" class=\"aligncenter size-full wp-image-2993\" \/><\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/div>\n<p style=\"margin:20px\">\n<hr style=\"height:2px;background-color:#8B5A2B;  margin-top: 6px; margin-bottom: 6px;border: 0 none;\"\/>\n<hr style=\"height:1px;background-color:#8B5A2B;  margin-top: 6px; margin-bottom: 6px;border: 0 none;\"\/>\n<p style=\"text-align: right;\"><span style=\"color: #8B5A2B;\"><strong>Equipe COM &#8211; OBMEP<\/strong><br \/>\n<\/span><\/p>\n<\/div>\n<p class=\"fim\"> <a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=33591\" title=\"Voltar.\" target=\"_blank\"><span style=\"color: #d66a1d;\">Voltar para p\u00e1gina inicial.<\/span><\/a><\/p>\n<p style=\"margin:20px\">\n<div style=\"text-align:center;\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width:487px;\">\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=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"background-color: #CDBA96; width:360px;\"> Voc\u00ea tamb\u00e9m ficou curioso?<br \/>\nEnt\u00e3o acompanhe a discuss\u00e3o da <a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=36131\" title=\"Visite a Sala!\" target=\"_blank\"><span style=\"color: #F5F5DC; font-size: 14px; font-family: 'Gill Sans', 'Gill Sans MT', Calibri, sans-serif;\"> <strong>pr\u00f3xima Sala<\/strong> <\/span><\/a><\/p>\n<p style=\"margin:20px\">\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Um pouco sobre pol\u00edgonos Os applets disponibilizados nesta p\u00e1gina poder\u00e3o ser utilizados on-line e off-line Para utiliz\u00e1-los off-line, basta copiar os arquivos disponibilizados abaixo de cada applet e abri-los no GeoGebra do seu computador ou tablet. Se voc\u00ea n\u00e3o tem o GeoGebra instalado, visite este site e procure pela vers\u00e3o adequada ao seu dispositivo. Estrelas, &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/um-pouco-sobre-poligonos-poligonos-uma-primeira-definicao-2\/um-pouco-sobre-poligonos-estrelas\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":33591,"menu_order":1,"comment_status":"open","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-27971","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/27971","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=27971"}],"version-history":[{"count":154,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/27971\/revisions"}],"predecessor-version":[{"id":181182,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/27971\/revisions\/181182"}],"up":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/33591"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=27971"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}