{"id":6916,"date":"2013-07-14T01:20:04","date_gmt":"2013-07-14T04:20:04","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=6916"},"modified":"2024-11-26T23:10:53","modified_gmt":"2024-11-27T02:10:53","slug":"brincando-com-geometria-mediatriz-de-um-segmento","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/brincando-com-geometria-mediatriz-de-um-segmento\/","title":{"rendered":".Brincando com Geometria: Mediatriz de um segmento"},"content":{"rendered":"<p style=\"margin:10px\">\n<hr style=\"height:1px;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:center;line-height:25px;margin:10px\"\">\n<span style=\"color: #8B5A2B; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 30px; line-height: 55px;\"><strong>A mediatriz de um segmento de reta <\/strong><\/span><\/p>\n<hr style=\"height:1px;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=\"margin:50px\">\n<div style=\"border: 8px double #8B5A2B; padding: 5px; padding-left: 7px; padding-right: 7px;\">\n<div style=\"text-align:center;\">\n<table style=\" height:auto; margin-left: auto; margin-right: auto;width:423px;\">\n<tr>\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 style=\"text-align: justify; border-style: dotted; border-color: #1C1C1C; border-width: thin; width: 300px; background-color: #d2b48c; padding: 15px;\"><span style=\"color: #1c1c1c; font-family: calibri, sans-serif; font-size: 17px;\">O que \u00e9 mesmo a mediatriz de um segmento?<\/span><\/p>\n<\/td>\n<\/tr>\n<\/table>\n<hr style=\"height:3px; width:98%;background-color:#8B5A2B; border: 0 none\"\/>\n<hr style=\"height:2px; width:90%;background-color:#8B5A2B; border: 0 none\"\/>\n<hr style=\"height:1px; width:82%;background-color:#8B5A2B; border: 0 none\"\/>\n<div style=\"border: 1px solid #000080; width: 650px; height: 350px; background-color: #d2b48c; margin-left: auto; margin-right: auto;\">\n<p style=\"margin:20px\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-6919\" style=\"border: 0px;\" alt=\"med22\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med22.png\" width=\"650\" height=\"291\" \/> <\/p>\n<hr style=\"height:3px; width:98%;background-color:#8B5A2B; border: 0 none\"\/><\/div>\n<hr style=\"height:1px; width:82%;background-color:#8B5A2B; border: 0 none\"\/>\n<hr style=\"height:2px; width:90%;background-color:#8B5A2B; border: 0 none\"\/>\n<hr style=\"height:3px; width:98%;background-color:#8B5A2B; border: 0 none\"\/>\n<p style=\"margin:20px\">\n<table style=\" height:auto; margin-left: auto; margin-right: auto; width:455px;\">\n<tr>\n<td style=\"border: 1px solid  #ffffff;\">\n<p style=\"text-align: justify; border-style: dotted; border-color: #1C1C1C; border-width: thin; width: 300px; background-color: #d2b48c; padding: 10px;\"><span style=\"color: #1c1c1c; font-family: calibri, sans-serif; font-size: 17px;\">Mas afinal de contas, uma mediatriz \u00e9 uma reta ou um lugar geom\u00e9trico?<\/span><\/p>\n<\/td>\n<td style=\"border: 1px solid #ffffff\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-full wp-image-2992\" style=\"border: 0px;\" alt=\"Probleminha2\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/02\/Probleminha2.png\" width=\"125\" height=\"135\" \/><\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto; margin-left: auto; margin-right: auto; width:530px;\">\n<tr>\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 style=\"text-align: center; border-style: dotted; border-color: #1C1C1C; border-width: thin; width: 400px; background-color: #d2b48c; padding: 10px;\"><span style=\"color: #1c1c1c; font-family: calibri, sans-serif; font-size: 17px;\">Por mais estranho que pare\u00e7a, as duas coisas &nbsp;.&nbsp;.&nbsp;.<\/span><\/p>\n<\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto; margin-left: auto; margin-right: auto;width:450px;\">\n<tr>\n<td style=\"border: 1px solid  #ffffff;\">\n<p style=\"text-align: justify; border-style: dotted; border-color: #1C1C1C; border-width: thin; width: 300px;  height:50px; background-color: #d2b48c; padding: 15px;\"><span style=\"color: #1c1c1c; font-family: calibri, sans-serif; font-size: 17px;\">Como assim ?&nbsp;!&nbsp;?&nbsp;!&nbsp;?&nbsp;!&nbsp;?&nbsp;!&nbsp;?<\/span><\/p>\n<\/td>\n<td style=\"border: 1px solid #ffffff\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-full wp-image-2993\" style=\"border: 0px;\" alt=\"Desafio2\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/02\/Desafio2.png\" width=\"120\" height=\"120\" \/><\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto; margin-left: auto; margin-right: auto;  width:370px;\">\n<tr>\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 style=\"text-align: justify; border-style: dotted; border-color: #1C1C1C; border-width: thin; width: 250px; background-color: #d2b48c; padding: 10px;\"><span style=\"color: #1c1c1c; font-family: calibri, sans-serif; font-size: 17px;\">Calma, vou tentar explicar. . .<\/span><\/p>\n<\/td>\n<\/tr>\n<\/table>\n<hr style=\"height:3px; width:98%;background-color:#8B5A2B; margin-top: 6px; margin-bottom: 6px;border: 0 none\"\/>\n<hr style=\"height:3px; width:98%;background-color:#8B5A2B; margin-top: 6px; margin-bottom: 6px;border: 0 none\"\/>\n<table style=\" height:auto; margin-left: auto; margin-right: auto;  width:640px;\">\n<tr>\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 style=\"text-align: justify; border-style: dotted; border-color: #1C1C1C; border-width: thin; width:520px; background-color: #d2b48c; padding: 10px;\"><span style=\"color: #1c1c1c; font-family: calibri, sans-serif; font-size: 17px;\"> Na planilha abaixo, fixamos um segmento [tex]\\overline{AB}[\/tex] e tra\u00e7amos a reta perpendicular a esse segmento e que passa pelo seu ponto m\u00e9dio (a reta que definimos como mediatriz). Com essa planilha voc\u00ea poder\u00e1 observar que os pontos dessa reta s\u00e3o equidistantes dos pontos [tex] A[\/tex] e [tex]B[\/tex].<\/span><\/p>\n<\/td>\n<\/tr>\n<\/table>\n<p style=\"margin:20px \">\n<div style=\"text-align:center;\">\n<input   value=\"Mostrar a Planilha\" style=\"cursor:pointer; padding: 0px; width: 180px; height: 40px; background-color:#CDBA96; 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 a Planilha';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Mostrar a Planilha';}\"    type=\"button\" \/>    <\/p>\n<div style=\"display: none;\">\n<div style=\"border: 1px dotted #8B5A2B; padding: 10px;padding-right: 15px; padding-left: 15px;  width: 85%;margin-left: auto; margin-right: auto; \">\n<p style=\"text-align: justify\"><span style=\"font-family: verdana, Arial, Helvetica, sans-serif; font-size: 14px; text-align: justify; color: #616161; line-height: 25px;\">1)  Abra a planilha  e aguarde o arquivo carregar.<br \/>\n2) Na planilha, movimente o ponto [tex]P[\/tex] sobre a reta perpendicular ao segmento [tex]\\overline{AB}[\/tex] e que passa pelo seu ponto m\u00e9dio. Observe que, independentemente da posi\u00e7\u00e3o que [tex]P[\/tex] ocupa sobre a reta, [tex]d(P,A)=d(P,B)[\/tex]. <\/span><\/p>\n<p style=\"margin:20px\">\n<p>\tPara abrir a planilha din\u00e2mica, clique <a href=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/Ykmf9Zih\/width\/530\/height\/707\/border\/ffffff\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/true\/ld\/false\/sri\/false\/at\/auto\" target=\"ap1\"><span style=\"color: #8B5A2B;font-size: 17px;\"><strong>AQUI<\/strong><\/span><\/a><br \/>\n<iframe  name=\"ap1\" width=\"530\" height=\"707\" style=\"border: 4px solid #8B5A2B;\"> <\/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: 14px; text-align: justify; color: #8B5A2B; line-height: 15px;\">Observamos que a planilha ajuda na visualiza\u00e7\u00e3o do resultado; mas, matematicamente, n\u00e3o substitui sua demonstra\u00e7\u00e3o.<\/span>\n\t<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px \">\n<hr style=\"height:3px; width:99%;background-color:#8B5A2B; border: 0 none\"\/>\n<table style=\" height:auto; margin-left: auto; margin-right: auto; width:640px;\">\n<tr>\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 style=\"text-align: justify; border-style: dotted; border-color: #1C1C1C; border-width: thin; width: 520px; background-color: #d2b48c; padding: 10px;\"><span style=\"color: #1c1c1c; font-family: calibri, sans-serif; font-size: 17px;\">Na pr\u00f3xima planilha, fixamos um segmento [tex]\\overline{AB}[\/tex] e tra\u00e7amos a reta perpendicular a esse segmento e que passa pelo ponto m\u00e9dio do segmento (a reta que definimos como mediatriz). Definimos pares de circunfer\u00eancias de centros em [tex]A[\/tex] e [tex]B[\/tex], ambas com raios iguais (variando de [tex]3,2 \\, [\/tex]cm a [tex]8 \\, [\/tex]cm), de modo a obtermos pontos equidistantes de [tex]A[\/tex] e de [tex]B[\/tex]: com efeito, se duas circunfer\u00eancias t\u00eam o mesmo raio, os pontos de interse\u00e7\u00e3o delas, quando existirem, s\u00e3o equidistantes de seus centros, no nosso caso, os pontos [tex]A[\/tex] e [tex]B[\/tex].<br \/>\nO que poderemos observar \u00e9 que todos os pontos de interse\u00e7\u00e3o determinados pertencem \u00e0 reta definida como a mediatriz do segmento.<br \/>\n<\/span><\/p>\n<\/td>\n<\/tr>\n<\/table>\n<p style=\"margin:30px \">\n<div style=\"text-align:center;\">\n<input   value=\"Mostrar a Planilha\" style=\"cursor:pointer; padding: 0px; width: 180px; height: 40px; background-color:#CDBA96; 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 a Planilha';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Mostrar a Planilha';}\"    type=\"button\" \/>    <\/p>\n<div style=\"display: none;\">\n<div style=\"border: 1px dotted #8B5A2B; padding: 10px;padding-right: 15px; padding-left: 15px;  width: 80%; margin-left: auto; margin-right: auto;\">\n<p style=\"text-align: justify\"><span style=\"font-family: verdana, Arial, Helvetica, sans-serif; font-size: 14px; text-align: justify; color: #616161; line-height: 25px;\">1)  Abra a planilha  e aguarde o arquivo carregar.<br \/>\n2) Na planilha, movimente horizontalmente o ponto vermelho r para modificar a medida do raio e obter v\u00e1rias circunfer\u00eancias.<\/span><\/p>\n<p style=\"margin:20px\">\n<p>Para abrir a planilha din\u00e2mica, clique <a href=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/UXzZ6QTU\/width\/574\/height\/705\/border\/ffffff\/smb\/false\/stb\/false\/stbh\/false\/ai\/false\/asb\/false\/sri\/false\/rc\/false\/ld\/false\/sdz\/false\/ctl\/false\" target=\"ap2\"><span style=\"color: #8B5A2B;font-size: 17px;\"><strong>AQUI<\/strong><\/span><\/a><br \/>\n<iframe  name=\"ap2\"  width=\"580\" height=\"708\" style=\"border: 4px solid #8B5A2B;\"> <\/iframe><span style=\"font-size: 14px;\"><br \/>\nOBMEP_ 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: 14px; text-align: justify; color: #8B5A2B; line-height: 15px;\">Observamos que a planilha ajuda na visualiza\u00e7\u00e3o do resultado; mas, matematicamente, n\u00e3o substitui sua demonstra\u00e7\u00e3o.<\/span>\n\t<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px \">\n<hr style=\"height:3px; width:98%;background-color:#8B5A2B; margin-top: 6px; margin-bottom: 6px;border: 0 none\"\/>\n<hr style=\"height:3px; width:98%;background-color:#8B5A2B; margin-top: 6px; margin-bottom: 6px;border: 0 none\"\/>\n<p style=\"margin:20px \">\n<table style=\" height:auto; margin-left: auto; margin-right: auto; width:610px;\">\n<tr>\n<td style=\"border: 1px solid  #ffffff;\">\n<p style=\"text-align: justify; border-style: dotted; border-color: #1C1C1C; border-width: thin; width: 500px; background-color: #d2b48c; padding: 15px;\"><span style=\"color: #1c1c1c; font-family: calibri, sans-serif; font-size: 17px;\">Mais uma coisa: o que significa &#8220;<em>A planilha ajuda na visualiza\u00e7\u00e3o do resultado; mas, matematicamente, n\u00e3o substitui sua demonstra\u00e7\u00e3o<\/em>&#8220;?<\/span><\/p>\n<\/td>\n<td style=\"border: 1px solid #ffffff\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-full wp-image-3060\" style=\"border: 0px;\" alt=\"Probleminha2\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/02\/carinha4.jpg\" width=\"90\" height=\"85\" \/> <\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto; margin-left: auto; margin-right: auto; width:640px;\">\n<tr>\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 style=\"text-align: justify; border-style: dotted; border-color: #1C1C1C; border-width: thin; width: 520px; background-color: #d2b48c; padding: 15px;\"><span style=\"color: #1c1c1c; font-family: calibri, sans-serif; font-size: 17px;\">Com as planilhas podemos verificar que os pontos da reta que definimos como mediatriz do segmento [tex]\\overline{AB}[\/tex] s\u00e3o equidistantes dos pontos [tex]A[\/tex] e [tex]B[\/tex] e, reciprocamente, que os pontos que s\u00e3o equidistantes de [tex]A[\/tex] e [tex]B[\/tex] pertencem \u00e0 reta que definimos como mediatriz do segmento [tex]\\overline{AB}[\/tex] APENAS para alguns pontos de um segmento particular [tex]\\overline{AB}[\/tex]. Mas n\u00e3o h\u00e1 como fazer essa verifica\u00e7\u00e3o em todas as situa\u00e7\u00f5es poss\u00edveis de pontos e segmentos. Assim temos que fazer uma justificativa gen\u00e9rica, ou seja, fazer o que chamamos na Matem\u00e1tica de demonstra\u00e7\u00e3o.<\/span><\/p>\n<\/td>\n<\/tr>\n<\/table>\n<p style=\"margin:20px \">\n<table style=\"text-align: justify; width: 99%;\">\n<tr>\n<td style=\"border: 5px double #D2B48C; background-color: #fffafa; padding-left: 20px; padding-right: 20px; color: #1c1c1c; font-family: verdana, Arial, Helvetica, sans-serif; font-size: 15px; line-height: 20px;\">\n<p style=\"text-align:center;\"><span style=\"color: #D2B48C; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 20px;\"><br \/>\n<strong>A demonstra\u00e7\u00e3o:<\/strong><\/span><\/p>\n<p><span style=\"color: #1c1c1c; font-family: verdana, Arial, Helvetica, sans-serif; font-size: 14px;line-height: 20px;\"> Fixe um plano [tex]\\alpha[\/tex] e considere nesse plano um segmento de reta [tex]\\overline{AB}[\/tex] cujo ponto m\u00e9dio denotaremos por [tex]M[\/tex]. Vamos mostrar que a reta perpendicular a esse segmento e que passa pelo seu ponto m\u00e9dio \u00e9 o lugar geom\u00e9trico dos pontos do plano [tex]\\alpha[\/tex] que equidistam de [tex]A[\/tex] e de [tex]B[\/tex].<br \/>\nComo um lugar geom\u00e9trico \u00e9 um conjunto de pontos, tal como uma reta, estamos propondo mostrar que, fixado um plano [tex]\\alpha[\/tex], dois conjuntos de pontos desse plano s\u00e3o iguais. Para isso, devemos mostrar que todo ponto de cada um dos conjuntos \u00e9 tamb\u00e9m ponto do outro.<br \/>\nDessa forma, fixemos um plano [tex]\\alpha[\/tex], fixemos dois pontos distintos desse plano, [tex]A[\/tex] e [tex]B[\/tex], e consideremos os seguintes conjuntos:<br \/>\n&nbsp;<br \/>\n[tex]\\qquad m:[\/tex] reta do plano [tex]\\alpha[\/tex] que \u00e9 perpendicular ao segmento [tex]\\overline{AB}[\/tex] e que passa pelo ponto m\u00e9dio [tex] M [\/tex] desse segmento;<br \/>\n&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(a reta que definimos como mediatriz desse segmento)<br \/>\n&nbsp;<br \/>\n[tex]\\qquad l[\/tex]: conjunto dos pontos do plano [tex]\\alpha[\/tex] que equidistam de [tex]A[\/tex] e [tex]B[\/tex].<br \/>\n&nbsp;<br \/>\nMostraremos que os conjuntos [tex]m[\/tex] e [tex]l[\/tex] s\u00e3o iguais, isso \u00e9, mostraremos que:<br \/>\n&nbsp;<br \/>\n&nbsp;&nbsp;&nbsp;&nbsp;(i) todo elemento de [tex]m[\/tex] \u00e9 elemento de [tex]l[\/tex] e<br \/>\n&nbsp;<br \/>\n&nbsp;&nbsp;&nbsp;&nbsp;(ii) todo elemento de [tex]l[\/tex] \u00e9 elemento de [tex]m[\/tex].<br \/>\n&nbsp;<br \/>\nVamos l\u00e1:<\/span><\/p>\n<p style=\"padding-left: 40px;\"><span style=\"color: #1c1c1c; font-family: verdana, Arial, Helvetica, sans-serif; line-height: 20px;\"> <strong>(i)<\/strong> Tomemos, inicialmente, um ponto [tex]P[\/tex] qualquer da reta [tex]m[\/tex].<br \/>\n[tex]\\bullet[\/tex] Se [tex]P=M[\/tex], ent\u00e3o [tex]P[\/tex] \u00e9 o ponto m\u00e9dio de [tex]\\overline{AB}[\/tex]. Assim, por defini\u00e7\u00e3o de ponto m\u00e9dio, [tex]d(P,A)=d(P,B)[\/tex], logo [tex]P \\in l[\/tex].<br \/>\n[tex]\\bullet[\/tex] Consideremos, agora, que [tex]P[\/tex] n\u00e3o seja o ponto m\u00e9dio de [tex]\\overline{AB}[\/tex], isto \u00e9, que [tex]P\\neq M [\/tex].<br \/>\nTracemos os segmentos [tex]\\overline{PA}[\/tex] e [tex]\\overline{PB}[\/tex] e consideremos os seguintes tri\u00e2ngulos: [tex]\\Delta PAM[\/tex] e [tex]\\Delta PBM[\/tex].<br \/>\nObservamos que os tri\u00e2ngulos definidos s\u00e3o congruentes, pelo <strong>caso lado-\u00e2ngulo-lado<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med4.png\" style=\"border: 0px;\" alt=\"med4\" width=\"447\" height=\"309\" class=\"aligncenter size-full wp-image-6982\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med4.png 447w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med4-300x207.png 300w\" sizes=\"auto, (max-width: 447px) 100vw, 447px\" \/><\/p>\n<p>Como [tex]\\Delta PAM \\cong \\Delta PBM[\/tex], em particular temos que os segmentos [tex]\\overline{PA}[\/tex] e [tex]\\overline{PB}[\/tex] s\u00e3o congruentes.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med5.png\" style=\"border: 0px;\" alt=\"med5\" width=\"426\" height=\"245\" class=\"aligncenter size-full wp-image-6983\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med5.png 426w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med5-300x172.png 300w\" sizes=\"auto, (max-width: 426px) 100vw, 426px\" \/><\/p>\n<p>Dessa forma, [tex]d(P,A)=d(P,B)[\/tex], donde temos que [tex]P \\in l[\/tex].<\/p>\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<strong>Pelo exposto, todo ponto de [tex]m[\/tex] \u00e9 ponto de [tex]l[\/tex] ([tex]m \\subset l[\/tex])<\/strong>.<br \/>\n&nbsp;<br \/>\n<strong>(ii)<\/strong> Tomemos, agora, um ponto [tex]P[\/tex] qualquer de [tex]l[\/tex]; assim [tex]P[\/tex] equidista dos pontos [tex]A[\/tex] e [tex]B[\/tex], isto \u00e9, [tex]d(P,A)=d(P,B)[\/tex].<br \/>\n[tex]\\bullet[\/tex] Se [tex]P[\/tex] for o ponto m\u00e9dio do segmento [tex]\\overline{AB}[\/tex], ent\u00e3o, necessariamente, [tex]P \\in m[\/tex], pela defini\u00e7\u00e3o da reta [tex]m[\/tex].<br \/>\n[tex]\\bullet[\/tex] Suponhamos que [tex]P[\/tex] n\u00e3o seja o ponto m\u00e9dio do segmento [tex]\\overline{AB}[\/tex] e tracemos os segmentos [tex]\\overline{PA}[\/tex] e [tex]\\overline{PB}[\/tex].<br \/>\nPodemos observar que os tri\u00e2ngulos [tex]\\Delta PAM[\/tex] e [tex]\\Delta PBM[\/tex] s\u00e3o congruentes, pelo <strong>caso lado-lado-lado<\/strong>,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med6.png\" style=\"border: 0px;\" alt=\"med6\" width=\"455\" height=\"248\" class=\"aligncenter size-full wp-image-6985\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med6.png 455w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med6-300x163.png 300w\" sizes=\"auto, (max-width: 455px) 100vw, 455px\" \/> <\/p>\n<p>e portanto os \u00e2ngulos [tex]\\angle PMA[\/tex] e [tex]\\angle PMB[\/tex] s\u00e3o congruentes, isto \u00e9, t\u00eam a mesma medida: [tex]m\\angle PMA = m\\angle PMB[\/tex].<br \/>\nMas esses \u00e2ngulos s\u00e3o suplementares, assim [tex]m\\angle PMA + m\\angle PMB=180^{\\circ}[\/tex], donde [tex]m\\angle PMA = m\\angle PMB=90^{\\circ}[\/tex].<br \/>\nDessa forma, observamos duas caracter\u00edsticas da reta  definida pelos pontos [tex]P[\/tex] e [tex]M[\/tex], aqui denominada [tex]r[\/tex]:<br \/>\n[tex]\\diamond \\,  \\,  \\,  \\,  M \\in r[\/tex]<br \/>\n[tex]\\diamond \\,  \\,  \\,  \\, r[\/tex] \u00e9 perpendicular ao segmento [tex]\\overline{AB}[\/tex]<br \/>\nPortanto a reta [tex]r[\/tex] determinada pelos pontos [tex]P[\/tex] e [tex]M[\/tex] \u00e9 a pr\u00f3pria mediatriz do segmento [tex]\\overline{AB}[\/tex] e, consequentemente, [tex]P \\in m[\/tex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med9.png\" style=\"border: 0px;\" alt=\"med9\" width=\"429\" height=\"323\" class=\"aligncenter size-full wp-image-6986\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med9.png 429w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/med9-300x225.png 300w\" sizes=\"auto, (max-width: 429px) 100vw, 429px\" \/><\/p>\n<p><strong>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Dessa forma, todo ponto de [tex]l[\/tex] \u00e9 tamb\u00e9m ponto de [tex]m[\/tex] ([tex]l \\subset m[\/tex]).<\/strong><\/p>\n<p style=\"margin:20px \">\n<p><strong>Finalmente, por (i) e por (ii) conclu\u00edmos que [tex]m = l[\/tex].<\/strong><\/span><\/p>\n<\/td>\n<\/tr>\n<\/table>\n<p style=\"margin:30px \">\n<table style=\" height:auto; margin-left: auto; margin-right: auto; width:640px;\">\n<tr>\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 style=\"text-align: justify; border-style: dotted; border-color: #1C1C1C; border-width: thin; width: 520px; background-color: #d2b48c; padding: 15px;\"><span style=\"color: #1c1c1c; font-family: calibri, sans-serif; font-size: 17px;\">Dessa forma, fixado um plano [tex]\\alpha[\/tex], podemos definir a mediatriz de um segmento como &#8220;a reta perpendicular ao segmento e que passa pelo seu ponto m\u00e9dio&#8221; ou como &#8220;o lugar geom\u00e9trico dos pontos do plano equidistantes das extremidades desse segmento&#8221;.<\/span><\/p>\n<\/td>\n<\/tr>\n<\/table>\n<table style=\" height:auto; margin-left: auto; margin-right: auto; width:279px;\">\n<tr>\n<td style=\"border: 1px solid  #ffffff;\">\n<p style=\"text-align: justify; border-style: dotted; border-color: #1C1C1C; border-width: thin; width: 150px; background-color: #d2b48c; padding: 15px;\"><span style=\"color: #1c1c1c; font-family: calibri, sans-serif; font-size: 17px;\">Agora sim !!!!<\/span><\/p>\n<\/td>\n<td style=\"border: 1px solid #ffffff\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/02\/carinha5.jpg\" style=\"border: 0px;\" alt=\"carinha5\" width=\"110\" height=\"101\" class=\"alignright size-full wp-image-3061\" \/><\/td>\n<\/tr>\n<\/table>\n<\/div>\n<p style=\"margin:30px \">\n<hr style=\"height:3px;background-color:#8B5A2B;margin-top: 6px; margin-bottom: 6px; border: 0 none\"\/>\n<hr style=\"height:2px; background-color:#8B5A2B; margin-top: 6px; margin-bottom: 10px; border: 0 none\"\/>\n<p style=\"text-align: right;\"><span style=\"color: #8B5A2B;\"><strong>Equipe COM &#8211; OBMEP<\/strong><\/p>\n<\/div>\n<p style=\"font-size: 16px;line-height: 30px;text-align: right; color: #8B5A2B;\">Setembro de 2025.<\/span><\/p>\n<table style=\" height:auto; margin-left: auto; margin-right: auto; width:530px; width:637px;\">\n<tr>\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 style=\"text-align: justify; border-style: dotted; border-color: #8B5A2B; border-width: thin; width: 500px; background-color: #f5f3f3; padding: 15px;\"><span style=\"color: #1C1C1C; font-family: calibri, sans-serif; font-size: 14px;\">Se for conveniente, voc\u00ea pode utilizar as planilhas din\u00e2micas <em>off-line<\/em>. Para isso, copie os arquivos abaixo e abra-os no GeoGebra do seu computador ou tablet.<br \/>\nMas n\u00e3o se esque\u00e7a de visitar <a href=\"https:\/\/www.geogebra.org\/download\" title=\"site do GeoGebra\" target=\"_blank\"><span style=\"color: #a20000; font-family: verdana, Arial, Helvetica, sans-serif; font-size: 12px;\"><strong>este site<\/strong><\/span><\/a> e fazer a instala\u00e7\u00e3o da vers\u00e3o do <em>GeoGebra <\/em>adequada ao dispositivo utilizado.<\/span><\/p>\n<\/td>\n<\/tr>\n<\/table>\n<div style=\" float:right;  width: 120px;\">\n<p style=\"text-align: center; border-style: solid; border-color :#a20000; border-width :thin; width:100px;  height:40px;  padding: 5px; background-color: #f5f3f3;\">\n<a href=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/07\/Med11.ggb\" title=\" Arquivo do GeoGebra\"  download=\"Med1.ggb\"><span style=\"color: #a20000; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 16px;\"><strong>Arquivo 1 <\/strong><\/span><\/a>\n <\/p>\n<p style=\"margin:20px;\">\n<p style=\"text-align: center; border-style: solid; border-color :#a20000; border-width :thin; width:100px;  height:40px; padding: 5px; background-color: #f5f3f3;\">\n<a href=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2018\/08\/mediatriz01.ggb\" title=\" Arquivo do GeoGebra\"  download=\"Med2.ggb\"><span style=\"color: #a20000; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 16px;\"><strong>Arquivo 2<\/strong><\/span><\/a>\n <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","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-6916","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/6916","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=6916"}],"version-history":[{"count":158,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/6916\/revisions"}],"predecessor-version":[{"id":223542,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/6916\/revisions\/223542"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=6916"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}