{"id":269,"date":"2023-07-10T01:42:26","date_gmt":"2023-07-10T01:42:26","guid":{"rendered":"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/"},"modified":"2023-07-10T01:42:26","modified_gmt":"2023-07-10T01:42:26","slug":"formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/","title":{"rendered":"F\u00f3rmula para dist\u00e2ncia entre dois pontos (geometria)"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea descobrir\u00e1 como calcular a dist\u00e2ncia entre dois pontos na geometria (f\u00f3rmula). Voc\u00ea tamb\u00e9m poder\u00e1 ver exemplos e, al\u00e9m disso, praticar com exerc\u00edcios resolvidos a dist\u00e2ncia entre dois pontos. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-104\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcual-es-la-formula-de-la-distancia-entre-dos-puntos\"><\/span> Qual \u00e9 a f\u00f3rmula para a dist\u00e2ncia entre dois pontos?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A dist\u00e2ncia entre dois pontos \u00e9 igual ao comprimento do segmento que os une. Portanto, em matem\u00e1tica, para determinar a dist\u00e2ncia entre dois pontos diferentes, devemos calcular os quadrados das diferen\u00e7as entre as suas coordenadas e depois encontrar a raiz da soma desses quadrados. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><\/div>\n<\/div>\n<p> Em outras palavras, a f\u00f3rmula utilizada para calcular a dist\u00e2ncia entre dois pontos diferentes no plano cartesiano \u00e9 a seguinte: <\/p>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> Considere as coordenadas de dois pontos distintos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-617b8c4c3becdf6f6f5ee619a5c8e9d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(x_1,y_1) \\qquad \\qquad B(x_2,y_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"209\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> A <strong>f\u00f3rmula para a dist\u00e2ncia entre dois pontos<\/strong> \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-db8fa011bbd9bef1ae4c02642919ea13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,B) = \\sqrt{(x_2-x_1)^2+(y_2-y_1)^2 \\vphantom{\\frac{1}{2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"277\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<\/div>\n<p> Esta f\u00f3rmula vem da magnitude de um vetor. Na verdade, o que estamos fazendo com esta f\u00f3rmula \u00e9 calcular a norma do vetor que \u00e9 determinada pelos dois pontos em quest\u00e3o. Voc\u00ea pode ler mais sobre isso na explica\u00e7\u00e3o do <a href=\"https:\/\/mathority.org\/pt\/modulo-de-uma-formula-vetorial-exemplos-de-exercicios-resolvidos\/\">que \u00e9 o m\u00f3dulo de um vetor<\/a> .<\/p>\n<p> Por outro lado, em geometria anal\u00edtica a demonstra\u00e7\u00e3o da f\u00f3rmula da dist\u00e2ncia entre dois pontos tamb\u00e9m pode ser feita atrav\u00e9s do teorema de Pit\u00e1goras: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-geometrique-distance-entre-deux-points.webp\" alt=\"f\u00f3rmula para a dist\u00e2ncia entre dois pontos\" class=\"wp-image-5012\" width=\"329\" height=\"329\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-106\"><\/div>\n<\/div>\n<p> O teorema de Pit\u00e1goras afirma que o quadrado da hipotenusa de um tri\u00e2ngulo ret\u00e2ngulo \u00e9 equivalente \u00e0 soma dos quadrados de seus catetos, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a030628d3e6320afdcc61fd4c3100ddd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl(d(A,B)\\bigr)^2 = (x_2-x_1)^2 + (y_2-y_1)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"280\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> E para obter a f\u00f3rmula basta encontrar a dist\u00e2ncia entre os 2 pontos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-db8fa011bbd9bef1ae4c02642919ea13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,B) = \\sqrt{(x_2-x_1)^2+(y_2-y_1)^2 \\vphantom{\\frac{1}{2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"277\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<p> Por fim, vale ressaltar que, se estiv\u00e9ssemos trabalhando com pontos de 3 coordenadas, a f\u00f3rmula da dist\u00e2ncia entre dois pontos no espa\u00e7o (em R3) seria a mesma, mas somando a coordenada Z: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-236a06c21ada3a9557cf73a29f47d8e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,B) = \\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2 \\vphantom{\\frac{1}{2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"374\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-como-calcular-la-distancia-entre-dos-puntos\"><\/span> Exemplo de c\u00e1lculo da dist\u00e2ncia entre dois pontos <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-109\"><\/div>\n<\/div>\n<p> Depois de vermos a defini\u00e7\u00e3o da f\u00f3rmula da dist\u00e2ncia entre dois pontos, vamos agora ver como determinar essa dist\u00e2ncia usando um exemplo:<\/p>\n<ul>\n<li> Encontre a dist\u00e2ncia entre os dois pontos a seguir:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cdf7bd1545cb6d04a4bb3851c6794466_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(-1,7) \\qquad \\qquad B(3,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"190\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para encontrar geometricamente a dist\u00e2ncia entre os dois pontos, basta aplicar a f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-db8fa011bbd9bef1ae4c02642919ea13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,B) = \\sqrt{(x_2-x_1)^2+(y_2-y_1)^2 \\vphantom{\\frac{1}{2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"277\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<p> Agora substitu\u00edmos as coordenadas dos pontos na f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02a4310cdfe1552b81816867261a17fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,B) = \\sqrt{(3-(-1))^2+(4-7)^2 \\vphantom{\\frac{1}{2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"271\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<p> E fazemos os c\u00e1lculos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e48600907e65f89fb9be0d55a2d3b3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} d(A,B)  &amp;= \\sqrt{(3+1)^2+(4-7)^2 \\vphantom{\\frac{1}{2}}} \\\\[2ex] &amp;= \\sqrt{4^2+(-3)^2 \\vphantom{\\frac{1}{2}}} \\\\[2ex] &amp;= \\sqrt{16+9}\\\\[2ex] &amp;= \\sqrt{25}\\\\[2ex] &amp; = \\bm{5}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"233\" width=\"244\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> A dist\u00e2ncia entre os dois pontos \u00e9, portanto, igual a 5 unidades.<\/p>\n<p> Obviamente, o valor da dist\u00e2ncia deve sempre dar-nos um sinal positivo, porque as dist\u00e2ncias s\u00e3o sempre positivas. Caso contr\u00e1rio, significa que cometemos um erro em uma etapa. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-distancia-entre-dos-puntos\"><\/span> Solu\u00e7\u00e3o de problemas de dist\u00e2ncia entre dois pontos<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Calcule a dist\u00e2ncia entre os dois pontos a seguir: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aa5be81901d5be989c5b72facdd42354_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(4,2) \\qquad \\qquad B(1,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"177\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para encontrar a dist\u00e2ncia geom\u00e9trica entre os dois pontos, basta usar a f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-db8fa011bbd9bef1ae4c02642919ea13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,B) = \\sqrt{(x_2-x_1)^2+(y_2-y_1)^2 \\vphantom{\\frac{1}{2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"277\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora substitu\u00edmos as coordenadas dos pontos na f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4afec52bf1fdc3c32a004a86e312b164_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,B) = \\sqrt{(1-4)^2+(5-2)^2 \\vphantom{\\frac{1}{2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"244\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E fazemos os c\u00e1lculos: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-81d70676cabdc2985f2ebe7b88c54e2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} d(A,B) &amp; = \\sqrt{(-3)^2+3^2 } \\\\[2ex] &amp; = \\sqrt{9+9 } \\\\[2ex] &amp; = \\bm{\\sqrt{18}} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"183\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 2<\/h3>\n<p> Encontre a dist\u00e2ncia entre os dois pontos a seguir: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9b2fed9d7325f254ded1553b488d7a0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(8,6) \\qquad \\qquad B(-4,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"190\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para encontrar a dist\u00e2ncia matem\u00e1tica entre os dois pontos, devemos usar a f\u00f3rmula correspondente:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-db8fa011bbd9bef1ae4c02642919ea13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,B) = \\sqrt{(x_2-x_1)^2+(y_2-y_1)^2 \\vphantom{\\frac{1}{2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"277\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora substitu\u00edmos as coordenadas dos pontos na f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5d1814bdae0e934c17e4d699ba2223c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,B) = \\sqrt{(-4-8)^2+(1-6)^2 \\vphantom{\\frac{1}{2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"258\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E fazemos os c\u00e1lculos: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1c9f9f5c93377868a352891d5b09630a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} d(A,B)  &amp;= \\sqrt{(-12)^2+(-5)^2 } \\\\[2ex] &amp;= \\sqrt{144+25 }\\\\[2ex] &amp;= \\sqrt{169} \\\\[2ex] &amp;= \\bm{13}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"149\" width=\"219\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 3<\/h3>\n<p> Calcule o per\u00edmetro do tri\u00e2ngulo formado pelos pontos A, B e C mostrados graficamente abaixo: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-distance-entre-deux-points.webp\" alt=\"exerc\u00edcio resolvido da dist\u00e2ncia entre dois pontos\" class=\"wp-image-3913\" width=\"338\" height=\"267\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro, precisamos identificar as coordenadas X e Y de cada ponto do gr\u00e1fico: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9a16d799fdc0fa3c371c35ba5f0f3a3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(2,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-df310f6e38e3e91250838800e5366e54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(4,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-68d3d057862c60995dc725a90e942df4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C(6,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora precisamos calcular a dist\u00e2ncia entre todos os pontos com a f\u00f3rmula: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1854b638ed0238a7b80632324e67ae0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,B) = \\sqrt{(4-2)^2+(4-1)^2 } = \\sqrt{4+9} =\\sqrt{13} = 3,61\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"435\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-741fa87cdc51782a0770cd384048b8fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,C) = \\sqrt{(6-2)^2+(2-1)^2 } = \\sqrt{16+1} =\\sqrt{17} = 4,12\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"443\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-79954ecf2f51f8c6e230b15c56dfd2f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(B,C) = \\sqrt{(6-4)^2+(2-4)^2 } = \\sqrt{4+4} =\\sqrt{8} = 2,83\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"427\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, o per\u00edmetro do tri\u00e2ngulo ser\u00e1 a soma dos comprimentos dos 3 lados: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9cdb85214a73df4e8a7e7ee238c1bc55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P=3,61+4,12+2,83= \\bm{10,56}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"251\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 4<\/h3>\n<p> Verifique se o tri\u00e2ngulo cujos v\u00e9rtices s\u00e3o os pontos A, B e C \u00e9 um tri\u00e2ngulo is\u00f3sceles. Mas os tr\u00eas pontos: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-df719532b223037ad6a672c50a294c3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(-4,5) \\qquad B(7,-5) \\qquad C(10,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"266\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para que um tri\u00e2ngulo seja is\u00f3sceles, dois de seus lados devem ser iguais. Devemos, portanto, encontrar o comprimento de cada um dos seus lados, que corresponde \u00e0s dist\u00e2ncias entre os seus v\u00e9rtices.<\/p>\n<p class=\"has-text-align-left\"> Portanto, calculamos a dist\u00e2ncia entre os v\u00e9rtices do tri\u00e2ngulo: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-786e56f8fe183e8143b1fb4f7f616226_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,B) = \\sqrt{(7-(-4))^2+(-5-5)^2 } = \\sqrt{11^2+(-10)^2} = \\sqrt{221}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"492\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-61cea14c48c09e4b4ae79b45d78cb746_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(A,C) = \\sqrt{(10-(-4))^2+(0-5)^2 } = \\sqrt{14^2+(-5)^2}  = \\sqrt{221}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"477\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-008513f83c089a4f9d4bc62b48a02aae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(B,C) = \\sqrt{(10-7)^2+(0-(-5))^2} = \\sqrt{3^2+5^2} = \\sqrt{34}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"430\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, o tri\u00e2ngulo tem 2 lados id\u00eanticos e o terceiro lado mede de forma diferente dos outros dois, ent\u00e3o \u00e9 efetivamente um tri\u00e2ngulo is\u00f3sceles.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 5<\/h3>\n<p> Encontre um ponto no eixo Y que seja equidistante dos dois pontos a seguir: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-94ff2ed3e9145831e9919ad74a4d4c4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(5,-3) \\qquad \\qquad B(-2,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"204\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Em primeiro lugar, se o ponto estiver localizado no eixo do computador (eixo OY), isso significa que a abcissa do ponto \u00e9 zero:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5cde05cd2f5f74f5290621fdbe275415_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(0,y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Segundo, se o ponto for equidistante dos pontos A e B, isso implica que a seguinte equa\u00e7\u00e3o \u00e9 satisfeita:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-26224760f49fac42cea9d39dfe53dd4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,A)= d(P,B)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, utilizando a f\u00f3rmula da dist\u00e2ncia entre dois pontos, podemos encontrar o valor da vari\u00e1vel <em>y<\/em> da equa\u00e7\u00e3o anterior:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a5f1368ddd79fe90319f21cc904d392_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{(5-0)^2+(-3-y)^2 \\vphantom{\\frac{1}{2}}}=\\sqrt{(-2-0)^2+(4-y)^2 \\vphantom{\\frac{1}{2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"374\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Como ambos os lados da equa\u00e7\u00e3o t\u00eam raiz, podemos simplific\u00e1-los:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-26c1085f06beb112f73bac79ec7a7a2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"5^2+(-3-y)^2=(-2)^2+(4-y)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"264\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resolvemos as pot\u00eancias e igualdades not\u00e1veis (ou produtos not\u00e1veis):<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59f4cfbad3f47e15ec8dd45d8eff5b0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"25+9+y^2+6y=4+16+y^2-8y\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"277\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E operamos at\u00e9 encontrarmos o valor da inc\u00f3gnita <em>y<\/em> : <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1e112c8e3a8ff239195ad850bd0e3f55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y^2+6y-y^2+8y=4+16-25-9\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"277\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc1582050e8015be81566a4ba06585ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"14y=-14\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67e3c61051e1e6f4d5ae83c0a7f8f51a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{-14}{14}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"75\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9f371b4e77462e28d9f6119571c92982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resumindo, o ponto que a declara\u00e7\u00e3o do problema nos perguntou \u00e9: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5d1dd065d80d8fce8e53869201ad7b2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{P(0,-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> Se voc\u00ea achou este artigo \u00fatil, provavelmente tamb\u00e9m se interessar\u00e1 por <a href=\"https:\/\/mathority.org\/pt\/distancia-entre-um-ponto-e-uma-formula-de-linha-exemplos-de-exercicios-resolvidos\/\">exerc\u00edcios sobre a dist\u00e2ncia entre um ponto e uma reta<\/a> . Na p\u00e1gina vinculada voc\u00ea encontrar\u00e1 n\u00e3o apenas exerc\u00edcios resolvidos passo a passo, mas tamb\u00e9m uma explica\u00e7\u00e3o detalhada do c\u00e1lculo da dist\u00e2ncia entre pontos e retas, exemplos e aplica\u00e7\u00e3o da f\u00f3rmula da dist\u00e2ncia entre um ponto e uma reta para determinar outro tipo de dist\u00e2ncia .<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea descobrir\u00e1 como calcular a dist\u00e2ncia entre dois pontos na geometria (f\u00f3rmula). Voc\u00ea tamb\u00e9m poder\u00e1 ver exemplos e, al\u00e9m disso, praticar com exerc\u00edcios resolvidos a dist\u00e2ncia entre dois pontos. Qual \u00e9 a f\u00f3rmula para a dist\u00e2ncia entre dois pontos? A dist\u00e2ncia entre dois pontos \u00e9 igual ao comprimento do segmento que os une. &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">F\u00f3rmula para dist\u00e2ncia entre dois pontos (geometria)<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[20],"tags":[],"class_list":["post-269","post","type-post","status-publish","format-standard","hentry","category-pontos-retas-e-planos"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>F\u00f3rmula para dist\u00e2ncia entre dois pontos (geometria) - Mathority<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"F\u00f3rmula para dist\u00e2ncia entre dois pontos (geometria) - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea descobrir\u00e1 como calcular a dist\u00e2ncia entre dois pontos na geometria (f\u00f3rmula). 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Mathority","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/","og_locale":"pt_BR","og_type":"article","og_title":"F\u00f3rmula para dist\u00e2ncia entre dois pontos (geometria) - Mathority","og_description":"Nesta p\u00e1gina voc\u00ea descobrir\u00e1 como calcular a dist\u00e2ncia entre dois pontos na geometria (f\u00f3rmula). Voc\u00ea tamb\u00e9m poder\u00e1 ver exemplos e, al\u00e9m disso, praticar com exerc\u00edcios resolvidos a dist\u00e2ncia entre dois pontos. Qual \u00e9 a f\u00f3rmula para a dist\u00e2ncia entre dois pontos? A dist\u00e2ncia entre dois pontos \u00e9 igual ao comprimento do segmento que os une. &hellip; F\u00f3rmula para dist\u00e2ncia entre dois pontos (geometria) Leia mais &raquo;","og_url":"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/","article_published_time":"2023-07-10T01:42:26+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-617b8c4c3becdf6f6f5ee619a5c8e9d7_l3.png"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"4 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"F\u00f3rmula para dist\u00e2ncia entre dois pontos (geometria)","datePublished":"2023-07-10T01:42:26+00:00","dateModified":"2023-07-10T01:42:26+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/"},"wordCount":897,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Pontos, retas e planos"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/","url":"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/","name":"F\u00f3rmula para dist\u00e2ncia entre dois pontos (geometria) - 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