{"id":72,"date":"2023-09-16T13:02:19","date_gmt":"2023-09-16T13:02:19","guid":{"rendered":"https:\/\/mathority.org\/pt\/exemplos-de-pontos-alinhados-exercicios-resolvidos\/"},"modified":"2023-09-16T13:02:19","modified_gmt":"2023-09-16T13:02:19","slug":"exemplos-de-pontos-alinhados-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/exemplos-de-pontos-alinhados-exercicios-resolvidos\/","title":{"rendered":"Pontos alinhados"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o do que s\u00e3o pontos alinhados. Voc\u00ea tamb\u00e9m ver\u00e1 todos os m\u00e9todos dispon\u00edveis para determinar se 3 (ou mais) pontos est\u00e3o alinhados. E mais, voc\u00ea encontrar\u00e1 diversos exemplos e at\u00e9 exerc\u00edcios resolvidos para praticar. <\/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%bfque-significa-que-los-puntos-estan-alineados\"><\/span> O que significa que os pontos est\u00e3o alinhados?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Na geometria anal\u00edtica, <strong>tr\u00eas ou mais pontos est\u00e3o alinhados<\/strong> se estiverem todos na mesma linha, ou seja, se puderem ser unidos tra\u00e7ando uma linha reta entre eles. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><\/div>\n<\/div>\n<p> Obviamente, 2 pontos estar\u00e3o sempre alinhados, pois voc\u00ea sempre pode tra\u00e7ar uma linha entre dois pontos. No entanto, tr\u00eas pontos n\u00e3o precisam estar na mesma linha. Principalmente, existem dois m\u00e9todos para saber se 3 ou mais pontos est\u00e3o alinhados:<\/p>\n<ul>\n<li> <strong>M\u00e9todo vetorial<\/strong> : consiste em ver se os vetores que formam os pontos s\u00e3o proporcionais.<\/li>\n<li> <strong>M\u00e9todo de equa\u00e7\u00e3o de linha<\/strong> : consiste em determinar se os pontos pertencem \u00e0 mesma linha.<\/li>\n<\/ul>\n<p> Abaixo est\u00e1 a explica\u00e7\u00e3o de cada procedimento e exemplos para que voc\u00ea decida qual \u00e9 o melhor para voc\u00ea. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-saber-si-3-o-mas-puntos-estan-alineados-con-el-metodo-de-los-vectores\"><\/span> Como saber se 3 (ou mais) pontos est\u00e3o alinhados com o m\u00e9todo vetorial <span class=\"ez-toc-section-end\"><\/span><\/h2>\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\"> Considerando 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-88818892831a9ada6aa3f369b1406d03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(x_1,y_1) \\quad B(x_2,y_2) \\quad C(x_3,y_3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"244\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Os tr\u00eas pontos est\u00e3o alinhados se os vetores<\/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-243865b783ec40e03ec861ac2ebcb279_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB}\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f10db290eb97b835b1c57091b3a33992_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{BC}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\"><\/p>\n<p> eles t\u00eam a mesma dire\u00e7\u00e3o, ou seja, se seus componentes s\u00e3o proporcionais. <\/p>\n<\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-106\"><\/div>\n<\/div>\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\/points-alignes-1.webp\" alt=\"pontos alinhados em um plano\" class=\"wp-image-1121\" width=\"267\" height=\"267\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Vejamos um exemplo de como isso \u00e9 feito:<\/p>\n<ul>\n<li> Determine se os tr\u00eas pontos a seguir est\u00e3o alinhados:<\/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-150343d1934b000b9bd8188f1418eb3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(1,2) \\quad B(4,4) \\quad C(-2,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"208\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Primeiro, calculamos os vetores entre os pontos. Basta calcular dois vetores diferentes:<\/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-987e7356216c577b32947f23519c407f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B - A = (4,4)- (1,2) = (3,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"287\" 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-ee1e4b4246ed129cc3f7adcb5315edc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{BC} = C-B =(-2,0)- (4,4)= (-6,-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"330\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E ent\u00e3o verificamos se as coordenadas dos vetores s\u00e3o proporcionais:<\/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-16253047910ee22fb08aebcdc967278f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-6}{3} = \\cfrac{-4}{2} = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"132\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Ao dividir as componentes X e as componentes Y dos dois vetores obtemos o mesmo resultado (-2), pois os vetores t\u00eam a mesma dire\u00e7\u00e3o e, portanto, <strong>os pontos est\u00e3o alinhados<\/strong> .<\/p>\n<p> Este m\u00e9todo tamb\u00e9m pode ser utilizado para saber se tr\u00eas ou mais pontos est\u00e3o alinhados no espa\u00e7o (em R3), bastando somar \u00e9 verificar se a terceira componente dos dois vetores (componente Z) tamb\u00e9m \u00e9 proporcional.<\/p>\n<p> Se este artigo for \u00fatil para voc\u00ea, provavelmente tamb\u00e9m ter\u00e1 interesse em saber como calcular o <a href=\"https:\/\/mathority.org\/pt\/formula-para-o-ponto-medio-de-um-vetor-de-segmento\/\">ponto m\u00e9dio entre dois pontos<\/a> , pois, obviamente, encontrar o ponto m\u00e9dio de 2 pontos \u00e9 uma forma de determinar um terceiro ponto alinhado com os outros dois pontos. Voc\u00ea pode ver como isso \u00e9 feito na p\u00e1gina do link, al\u00e9m disso voc\u00ea tamb\u00e9m poder\u00e1 ver exemplos e exerc\u00edcios resolvidos passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-averiguar-si-3-o-mas-puntos-estan-alineados-con-el-metodo-de-la-ecuacion-de-la-recta\"><\/span> Como saber se 3 (ou mais) pontos est\u00e3o alinhados com o m\u00e9todo da equa\u00e7\u00e3o da linha reta <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> Como acabamos de ver na se\u00e7\u00e3o anterior, uma forma de estudar o alinhamento de 3 ou mais pontos \u00e9 utilizar os vetores que podem se formar entre eles. Bem, outro m\u00e9todo \u00e9 come\u00e7ar com a equa\u00e7\u00e3o de uma reta: <\/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\"> Considerando 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-88818892831a9ada6aa3f369b1406d03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(x_1,y_1) \\quad B(x_2,y_2) \\quad C(x_3,y_3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"244\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Os tr\u00eas pontos est\u00e3o alinhados se todos pertencerem \u00e0 mesma linha. Portanto, para saber se tr\u00eas ou mais pontos est\u00e3o alinhados, devem ser seguidos os seguintes passos:<\/p>\n<ol style=\"color:#42A5F5\">\n<li> <span style=\"color:#0c0c0c\">Encontre a equa\u00e7\u00e3o da reta que passa por dois dos tr\u00eas pontos.<\/span><\/li>\n<li> <span style=\"color:#0c0c0c\">Verifique se o terceiro ponto tamb\u00e9m pertence \u00e0 linha. Neste caso significa que os 3 pontos est\u00e3o alinhados, por\u00e9m se a condi\u00e7\u00e3o n\u00e3o for atendida significa que os pontos n\u00e3o est\u00e3o alinhados.<\/span> <\/li>\n<\/ol>\n<\/div>\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\/formule-de-3-points-alignes.webp\" alt=\"f\u00f3rmula de tr\u00eas pontos alinhados\" class=\"wp-image-1143\" width=\"275\" height=\"262\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Como exemplo, resolveremos um exerc\u00edcio utilizando este m\u00e9todo:<\/p>\n<ul>\n<li> Verifique se os tr\u00eas pontos a seguir est\u00e3o alinhados:<\/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-0955033d07a0ffa4311b6477fec3b153_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(3,1) \\quad B(1,4) \\quad C(5,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"208\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Em primeiro lugar, devemos calcular a equa\u00e7\u00e3o da reta que passa pelos pontos A e B. Portanto, encontramos o vetor diretor da reta:<\/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-4a80df873f9ac7ae3d132a551e8de7e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B-A = (1,4) - (3,1) = (-2,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"301\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Agora voc\u00ea tem que construir a equa\u00e7\u00e3o da reta, voc\u00ea pode escolher o tipo que deseja: param\u00e9trico, impl\u00edcito, geral, etc. Mas neste caso usaremos a equa\u00e7\u00e3o cont\u00ednua. Portanto, a equa\u00e7\u00e3o cont\u00ednua da reta que passa pelo ponto A e pelo ponto B ser\u00e1:<\/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-b6f94bd712f1b5eb3e086f90b1ab497a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x-3}{-2}=\\cfrac{y-1}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"106\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Assim que tivermos a equa\u00e7\u00e3o da reta, devemos verificar se o outro ponto tamb\u00e9m pertence \u00e0 mesma reta. Para fazer isso, substitu\u00edmos as coordenadas do ponto C na equa\u00e7\u00e3o da reta: <\/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-45717c1e198eb634f35329c7af91f1c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{5-3}{-2}=\\cfrac{-2-1}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"127\" 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-fb2241c675eff5beda831f375678af8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2}{-2}=\\cfrac{-3}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"79\" 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-962c6302723831d83dd3adb9955b8ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-1=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Empatamos, ent\u00e3o o ponto satisfaz a equa\u00e7\u00e3o da reta. E portanto <strong>os 3 pontos s\u00e3o colineares<\/strong> .<\/p>\n<p> Ressalta-se que um conjunto de pontos alinhados n\u00e3o precisa ser equidistante, ou seja, a dist\u00e2ncia entre v\u00e1rios pontos alinhados pode ser diferente. Voc\u00ea pode ver a diferen\u00e7a entre os dois conceitos na explica\u00e7\u00e3o da <a href=\"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/\">dist\u00e2ncia entre dois pontos (geometria)<\/a> , onde tamb\u00e9m pode ver exemplos e exerc\u00edcios resolvidos passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-puntos-alineados\"><\/span> Exerc\u00edcios resolvidos de pontos alinhados<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Determine se os tr\u00eas pontos a seguir est\u00e3o alinhados: <\/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-508fe1a58d29e8dd0e5f25e9c431e07e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(1,1) \\quad B(2,-1) \\quad C(-2,7)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"222\" 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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Podemos escolher um dos dois m\u00e9todos que vimos para resolver o problema. Neste caso usaremos o m\u00e9todo vetorial.<\/p>\n<p class=\"has-text-align-left\"> Primeiro, calculamos os vetores entre os 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-8a4c888c595453f36f08fd6efb79c879_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B - A = (2,-1)- (1,1) = (1,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"315\" 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-2ee5603a2cc587ab3dcffa296aeef0e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{BC} = C-B =(-2,7)- (2,-1)= (-4,8)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"330\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora verificamos se as coordenadas cartesianas dos vetores s\u00e3o proporcionais:<\/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-e78dec44294d4d784e5cf93ff4cf6fd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-4}{1} = \\cfrac{8}{-2} = -4\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"125\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ao dividir as componentes X e as componentes Y dos dois vetores entre si obtemos o mesmo resultado (-4), portanto os vetores t\u00eam a mesma dire\u00e7\u00e3o. Fato que indica que <strong>os pontos est\u00e3o alinhados<\/strong> .<\/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> Dados 3 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-a211b45658e3ecc9a6d99ab9c25cef6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(6,3) \\quad B(4,-3) \\quad C(4,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"208\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Determine quais deles se alinham com 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-e69edcc5f0f57a26e5113fc037b79a33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D(3,-1) \\quad E(2,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"139\" 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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste caso utilizaremos o m\u00e9todo da equa\u00e7\u00e3o da reta, assim salvaremos alguns c\u00e1lculos.<\/p>\n<p class=\"has-text-align-left\"> Calculamos, portanto, a equa\u00e7\u00e3o cont\u00ednua da reta que passa pelos pontos D e E: <\/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-b49904508e43c2689dbd43b72c54a343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{DE} = E-D = (2,1) - (3,-1) = (-1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"318\" 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-e2006fb32975727fc10dea3e135fcc1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x-3}{-1}=\\cfrac{y+1}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"106\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora vamos verificar quais pontos correspondem \u00e0 equa\u00e7\u00e3o da reta e, portanto, est\u00e3o alinhados com os pontos D e E, e quais n\u00e3o est\u00e3o.<\/p>\n<p class=\"has-text-align-left\"> Verificamos o ponto A: <\/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-1719d4a555050da2ea17f56d610e569e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{6-3}{-1}=\\cfrac{3+1}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"105\" 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-c9c8e342285cc2960cd1ac221202c854_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{3}{-1}=\\cfrac{4}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"57\" 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-661377ab0472015b8558320f2c01ef58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3\\neq 2\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"53\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A equa\u00e7\u00e3o da reta n\u00e3o \u00e9 verdadeira, ent\u00e3o <strong>o ponto A n\u00e3o est\u00e1 alinhado com os pontos D e E.<\/strong><\/p>\n<p class=\"has-text-align-left\"> Agora verificamos o ponto B: <\/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-c627d26517f60ddc4ac6921f8e8f3e07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{4-3}{-1}=\\cfrac{-3+1}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"127\" 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-edc9b8b5c0349b7f757d00cbe0dc7d02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{1}{-1}=\\cfrac{-2}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"79\" 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-9f4f670113c39f3a05ad405e92176773_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-1= -1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Neste caso a equa\u00e7\u00e3o da reta \u00e9 cumprida, ent\u00e3o <strong>o ponto B \u00e9 colinear com os pontos D e E.<\/strong><\/p>\n<p class=\"has-text-align-left\"> E por fim, repetimos o processo com o ponto C: <\/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-fca526dda7c61b4d49f377dfb3bddc3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{4-3}{-1}=\\cfrac{5+1}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"105\" 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-50dc4b51810a1541b1c647128826857f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{1}{-1}=\\cfrac{6}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"57\" 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-0967dbdc2483e19c1585ce93af3f75b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-1\\neq 3\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"54\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A equa\u00e7\u00e3o da reta n\u00e3o \u00e9 verdadeira, ent\u00e3o <strong>o ponto C n\u00e3o est\u00e1 alinhado com os pontos D e E.<\/strong><\/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> Encontrando o valor do desconhecido<\/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-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> para que os 3 pontos a seguir estejam alinhados: <\/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-3d24a8001da7ee9bc585c9cde8c495ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(k,5) \\quad B(-1,-4) \\quad C(k-1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"254\" 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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste caso usaremos o m\u00e9todo vetorial.<\/p>\n<p class=\"has-text-align-left\"> Tentamos, portanto, calcular os vetores entre os 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-b81dc7bc509c6e2589c62f78934129ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B - A = (-1,-4)- (k,5) = (-1-k,-9)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"375\" 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-e4c6260bf90e1ffb02aaf6dc7297cda1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{BC} = C-B =(k-1,2)- (-1,-4)= (k,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"348\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para que a colinearidade de tr\u00eas pontos seja satisfeita, as coordenadas dos dois vetores devem ser proporcionais. Portanto, aplicamos esta condi\u00e7\u00e3o:<\/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-8478d3909be370139f11503805517fcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-1-k}{k} = \\cfrac{-9}{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"119\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E resolvemos a equa\u00e7\u00e3o: <\/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-476a5e44e3e0eb3057a5cc26e27cb77b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1-k)\\cdot 6 = -9 \\cdot k\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" 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-a2a9cb146b7bb7de4a735e0c2a409207_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-6-6k = -9k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"118\" style=\"vertical-align: 0px;\"><\/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-9dedb7656b3248213103847e15e46388_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-6k+9k = 6\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"104\" style=\"vertical-align: -2px;\"><\/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-3eba160999a1f5f128b874b5d8d18a60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3k = 6\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\"><\/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-01f0a6048dc5d48632c2e10fa8179c47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k=\\cfrac{6}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"44\" 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-20b989bf2c21866deb6a2b220f87dfff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{k=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o para que os 3 pontos fiquem alinhados<\/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-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> deve valer 2.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o do que s\u00e3o pontos alinhados. Voc\u00ea tamb\u00e9m ver\u00e1 todos os m\u00e9todos dispon\u00edveis para determinar se 3 (ou mais) pontos est\u00e3o alinhados. E mais, voc\u00ea encontrar\u00e1 diversos exemplos e at\u00e9 exerc\u00edcios resolvidos para praticar. O que significa que os pontos est\u00e3o alinhados? Na geometria anal\u00edtica, tr\u00eas ou mais pontos est\u00e3o &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/exemplos-de-pontos-alinhados-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">Pontos alinhados<\/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-72","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>\u25b7 Como posso saber se 3 (ou mais) pontos est\u00e3o alinhados? 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