{"id":233,"date":"2023-07-10T19:30:40","date_gmt":"2023-07-10T19:30:40","guid":{"rendered":"https:\/\/mathority.org\/pt\/distancia-entre-um-ponto-e-uma-formula-de-linha-exemplos-de-exercicios-resolvidos\/"},"modified":"2023-07-10T19:30:40","modified_gmt":"2023-07-10T19:30:40","slug":"distancia-entre-um-ponto-e-uma-formula-de-linha-exemplos-de-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/distancia-entre-um-ponto-e-uma-formula-de-linha-exemplos-de-exercicios-resolvidos\/","title":{"rendered":"Dist\u00e2ncia entre um ponto e uma reta"},"content":{"rendered":"<p>Aqui voc\u00ea encontrar\u00e1 a f\u00f3rmula usada para calcular a dist\u00e2ncia entre um ponto e uma linha. Al\u00e9m disso, voc\u00ea poder\u00e1 ver diversos exemplos e exerc\u00edcios resolvidos de dist\u00e2ncias entre pontos e retas e, ainda, as aplica\u00e7\u00f5es que esta opera\u00e7\u00e3o possui (por exemplo, encontrar a dist\u00e2ncia entre retas paralelas). <\/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=\"formula-de-la-distancia-entre-un-punto-y-una-recta\"><\/span> F\u00f3rmula para a dist\u00e2ncia entre um ponto e uma linha<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A <strong>dist\u00e2ncia entre um ponto e uma linha<\/strong> \u00e9 a dist\u00e2ncia mais curta entre esse ponto e a linha. Matematicamente, esta dist\u00e2ncia m\u00ednima equivale ao comprimento do segmento tra\u00e7ado do ponto \u00e0 reta e que \u00e9 perpendicular \u00e0 reta. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><\/div>\n<\/div>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/distance-entre-un-point-et-une-droite-1.webp\" alt=\"qual \u00e9 a dist\u00e2ncia entre um ponto e uma reta\" class=\"wp-image-1951\" width=\"326\" height=\"291\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Depois de vermos o conceito geom\u00e9trico de dist\u00e2ncia entre um ponto e uma reta, vamos ver qual \u00e9 a f\u00f3rmula usada para calcular essa dist\u00e2ncia: <\/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\"> Dada a equa\u00e7\u00e3o impl\u00edcita (ou geral) de uma reta e as coordenadas de qualquer ponto de um plano:<\/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-127fb199bd8a31769e3814d2581888eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r: \\ Ax+By+C=0 \\qquad \\qquad P(p_x,p_y)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"306\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p style=\"text-align:left;\"> A <strong>f\u00f3rmula para a dist\u00e2ncia entre um ponto e uma linha<\/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-8064e6650ca06c9d921e13e956ab02a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert A\\cdot p_x + B\\cdot p_y +C\\rvert}{\\sqrt{A^2+B^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"232\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<\/div>\n<p> <strong>Importante:<\/strong> Observe que a equa\u00e7\u00e3o da reta na f\u00f3rmula est\u00e1 na forma de uma equa\u00e7\u00e3o impl\u00edcita (ou geral), portanto se tivermos a reta expressa em outro tipo de equa\u00e7\u00e3o, precisamos pass\u00e1-la primeiro para sua equa\u00e7\u00e3o impl\u00edcita e depois podemos aplicar a f\u00f3rmula. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-como-calcular-la-distancia-entre-un-punto-y-una-recta\"><\/span> Exemplo de c\u00e1lculo da dist\u00e2ncia entre um ponto e uma linha <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-106\"><\/div>\n<\/div>\n<p> Abaixo voc\u00ea pode ver um exemplo de c\u00e1lculo da dist\u00e2ncia entre um ponto e uma linha:<\/p>\n<ul>\n<li> Encontre a dist\u00e2ncia entre o ponto\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> e a lei<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c5986f031932e6b3512dc564514c34b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r:\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\"><\/p>\n<\/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-7ddc3d4b82281ca90fa991e092194056_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(2,-1) \\qquad \\qquad r: \\ 3x+4y-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"289\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para calcular a dist\u00e2ncia entre o ponto e a reta, basta aplicar sua 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-8064e6650ca06c9d921e13e956ab02a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert A\\cdot p_x + B\\cdot p_y +C\\rvert}{\\sqrt{A^2+B^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"232\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Agora substitu\u00edmos cada termo pelo seu valor:<\/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-5117783b26d70ca74c38e1c3f7c94505_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert 3\\cdot 2 + 4\\cdot (-1)-5\\rvert}{\\sqrt{3^2+4^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"228\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> E finalmente calculamos a dist\u00e2ncia: <\/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-d1ab18e8e237f777d2ae0dfbc41055a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert 6 -4-5\\rvert}{\\sqrt{9+16}} =\\cfrac{\\lvert -3 \\rvert}{\\sqrt{25}} = \\mathbf{\\cfrac{3}{5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"251\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"distancia-entre-dos-rectas-paralelas\"><\/span> Dist\u00e2ncia entre duas linhas paralelas <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> Uma das aplica\u00e7\u00f5es do c\u00e1lculo da dist\u00e2ncia entre uma reta e um ponto \u00e9 encontrar a dist\u00e2ncia entre retas paralelas.<\/p>\n<p> Obviamente, para entender o conceito que explicaremos a seguir, voc\u00ea deve saber <a href=\"https:\/\/mathority.org\/pt\/definicao-e-exemplos-de-linhas-paralelas\/\">o que s\u00e3o retas paralelas<\/a> , ent\u00e3o se voc\u00ea n\u00e3o sabe exatamente sua defini\u00e7\u00e3o, deixamos um link onde explicamos detalhadamente e voc\u00ea tamb\u00e9m poder\u00e1 ver exemplos de linhas paralelas.<\/p>\n<p> <strong>Para encontrar a dist\u00e2ncia entre duas retas paralelas, basta pegar um ponto em uma das duas retas e calcular a dist\u00e2ncia desse ponto at\u00e9 a outra reta.<\/strong> <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/distance-entre-un-point-et-une-ligne-en-ligne.webp\" alt=\"dist\u00e2ncia entre um ponto e uma reta\" class=\"wp-image-1960\" width=\"421\" height=\"358\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Assim, para determinar a dist\u00e2ncia entre duas retas paralelas, tamb\u00e9m \u00e9 utilizada a f\u00f3rmula da dist\u00e2ncia entre uma reta e um ponto.<\/p>\n<p> Por outro lado, se ao usar a f\u00f3rmula obtivermos uma dist\u00e2ncia de 0 unidades, isso significa que as retas se tocam em algum ponto e, portanto, as retas n\u00e3o s\u00e3o paralelas, mas se cruzam, coincidentes ou perpendiculares. Se quiser, voc\u00ea pode conferir as diferen\u00e7as entre esse tipo de linha em nosso site.<\/p>\n<p> Ent\u00e3o vamos ver como resolver um problema de dist\u00e2ncia entre duas retas paralelas atrav\u00e9s de um exemplo:<\/p>\n<ul>\n<li> Encontre a dist\u00e2ncia entre as duas linhas paralelas 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-8be694174fbd8330f207d16a9fb4bb89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r: \\ 2x-4y-6=0 \\qquad \\qquad s: \\ -x+2y+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"378\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> A primeira coisa que precisamos fazer \u00e9 marcar um ponto em uma das linhas (a que voc\u00ea deseja). Neste caso, calcularemos um ponto na 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-23a7daa116b8874af1538c91f8d239de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"12\" style=\"vertical-align: 0px;\"><\/p>\n<p> Para fazer isso, voc\u00ea deve atribuir um valor a uma das vari\u00e1veis, por exemplo faremos<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d762821a7c6da83f02380639f43ef8fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" 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-3b748efe6f89a8c847e2e6c2d5a78db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-x+2y+4 =0 \\ \\xrightarrow{x \\ = \\ 0} \\ -0+2y+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"321\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> E agora limpamos a outra vari\u00e1vel (<\/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-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> ) da equa\u00e7\u00e3o obtida para saber quanto vale neste momento: <\/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-91edb06d37339167f458f910de16d57b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2y=-4\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"65\" 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-9901e9a85409e4bcbfcbffdcf97cb175_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y= \\cfrac{-4}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"66\" 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-50b25376eef215b49997f236615b6d6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y= -2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Portanto, o ponto obtido 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-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> Leste:<\/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-a9176e7fde633e028a8349a5bc422e03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(0,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E como j\u00e1 temos um ponto em uma reta, calculamos a dist\u00e2ncia desse ponto at\u00e9 a outra reta usando 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-8064e6650ca06c9d921e13e956ab02a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert A\\cdot p_x + B\\cdot p_y +C\\rvert}{\\sqrt{A^2+B^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"232\" style=\"vertical-align: -16px;\"><\/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-451c7425d16dbf9c209df7a37777597b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert 2\\cdot 0 + (-4)\\cdot (-2) +(-6)\\rvert}{\\sqrt{2^2+(-4)^2}}= \\cfrac{2}{\\sqrt{20}}=\\bm{0,45}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"398\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-distancia-entre-un-punto-y-una-recta\"><\/span> Problemas resolvidos de dist\u00e2ncia entre um ponto e uma linha<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Calcular a dist\u00e2ncia entre o ponto<\/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-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> e a lei <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c5986f031932e6b3512dc564514c34b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r:\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" 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-15463bc0f037173b2283051b2e915dae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(4,2) \\qquad \\qquad r: \\ 5x-3y+6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"275\" 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 entre um ponto e uma reta, basta aplicar sua 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-8064e6650ca06c9d921e13e956ab02a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert A\\cdot p_x + B\\cdot p_y +C\\rvert}{\\sqrt{A^2+B^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"232\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Substitu\u00edmos cada termo pelo seu valor e calculamos a dist\u00e2ncia: <\/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-e840997220799ac7d54b6efffc91533a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert 5\\cdot 4 + (-3)\\cdot 2+6\\rvert}{\\sqrt{5^2+(-3)^2}}=\\cfrac{20}{\\sqrt{34}}= \\bm{3,43}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"344\" style=\"vertical-align: -20px;\"><\/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> Qual \u00e9 a dist\u00e2ncia entre o ponto<\/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-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> e a lei<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> ? <\/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-41ed4564d284382ca1818f75faf32b27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-3,-1) \\qquad \\qquad r: \\  y=-3x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"276\" 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\"> Neste caso, a equa\u00e7\u00e3o da reta est\u00e1 na forma impl\u00edcita (ou geral). Em vez disso, para usar a f\u00f3rmula da dist\u00e2ncia de um ponto a uma reta, a reta deve ser expressa como uma equa\u00e7\u00e3o impl\u00edcita. Devemos, portanto, primeiro transformar a reta e pass\u00e1-la para uma equa\u00e7\u00e3o impl\u00edcita (basta passar todos os termos do mesmo lado da 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-1e964a7afee927714fb86c7157d4b881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r: \\  3x+y-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"142\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E uma vez que a reta j\u00e1 est\u00e1 na forma expl\u00edcita, podemos agora usar a f\u00f3rmula para a dist\u00e2ncia entre um ponto e uma 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-8064e6650ca06c9d921e13e956ab02a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert A\\cdot p_x + B\\cdot p_y +C\\rvert}{\\sqrt{A^2+B^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"232\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, substitu\u00edmos cada termo pelo seu valor e calculamos a dist\u00e2ncia: <\/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-5da11b73842cb053431781741f29cf97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert 3\\cdot (-3) + 1\\cdot (-1)-5\\rvert}{\\sqrt{3^2+1^2}}=\\cfrac{\\lvert -9-1-5\\rvert}{\\sqrt{9+1}}=\\cfrac{15}{\\sqrt{10}}= \\bm{4,74}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"491\" style=\"vertical-align: -16px;\"><\/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> Qual \u00e9 a dist\u00e2ncia entre as duas linhas 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-9ad10fd8e20d413e433851810e823172_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r: \\ x+3y-4=0 \\qquad \\qquad s: \\ 2x+6y+6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"364\" style=\"vertical-align: -4px;\"><\/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\"> Primeiro, verificaremos se se trata de duas retas paralelas. Para isso, os coeficientes das vari\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-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" 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-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> devem ser proporcionais entre si, mas n\u00e3o aos termos independentes:<\/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-1a5385dff1a5c143885cac238927cea7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{1}{2} = \\cfrac{3}{6}\\neq \\cfrac{-4}{6} \\ \\longrightarrow \\ \\text{Paralelas}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"220\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Na verdade, as linhas s\u00e3o paralelas, podemos, portanto, aplicar o procedimento.<\/p>\n<p class=\"has-text-align-left\"> Agora precisamos pegar um ponto de uma das retas (a que voc\u00ea deseja). Neste caso, calcularemos um ponto na 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-23a7daa116b8874af1538c91f8d239de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"12\" style=\"vertical-align: 0px;\"><\/p>\n<p> Para fazer isso, voc\u00ea deve atribuir um valor a uma das vari\u00e1veis, por exemplo faremos<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d762821a7c6da83f02380639f43ef8fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" 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-cfbc09666c2be26abbecc34491f0f0a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x+6y+6=0 \\ \\xrightarrow{x \\ = \\ 0} \\ 2\\cdot 0+6y+6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"325\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora limpamos a outra vari\u00e1vel (<\/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-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> ) da equa\u00e7\u00e3o obtida para saber seu valor neste ponto: <\/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-d2147ae4d002aba7f27fda064a5f3d15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6y=-6\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"65\" 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-011e4abb1bbc3ef5f362695736e2b1f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y= \\cfrac{-6}{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"66\" 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-8330e0406bdbde3e65e8142fafceeee6_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\"> Para que o ponto obtido 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-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> Leste:<\/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-d4834bffe8509fbc113d7be09e378a7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(0,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Depois de conhecermos um ponto em uma reta, calculamos a dist\u00e2ncia desse ponto at\u00e9 a outra reta 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-8064e6650ca06c9d921e13e956ab02a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert A\\cdot p_x + B\\cdot p_y +C\\rvert}{\\sqrt{A^2+B^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"232\" style=\"vertical-align: -16px;\"><\/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-04bdc4f7bf4e33872b9a1dae673198df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert 1\\cdot 0 + 3\\cdot (-1) +(-4)\\rvert}{\\sqrt{1^2+3^2}}= \\cfrac{7}{\\sqrt{10}}=\\bm{2,21}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"371\" style=\"vertical-align: -16px;\"><\/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> Calcule o valor da inc\u00f3gnita<\/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> de modo que a dist\u00e2ncia entre o ponto<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> e a lei<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> ou seja, 5 unidades. <\/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-4a67915da76e50711bd1ab382c252d0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2,5) \\qquad \\qquad r: \\ 12x-5y+k=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"299\" 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\"> Devemos primeiro aplicar a f\u00f3rmula para a dist\u00e2ncia entre um ponto e uma 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-8064e6650ca06c9d921e13e956ab02a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert A\\cdot p_x + B\\cdot p_y +C\\rvert}{\\sqrt{A^2+B^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"232\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora substitu\u00edmos cada termo pelo seu valor e simplificamos a express\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-33ba5ddeb270550eb80a95a092204c53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,r)= \\cfrac{\\lvert 12\\cdot (-2) + (-5)\\cdot 5+k\\rvert}{\\sqrt{12^2+(-5)^2}}= \\cfrac{\\lvert -24-25+k\\rvert}{\\sqrt{169}}=\\cfrac{\\lvert -49+k\\rvert}{13}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"501\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O enunciado do problema nos diz que a dist\u00e2ncia entre o ponto e a reta deve ser igual a 5, ent\u00e3o igualamos a express\u00e3o anterior a 5:<\/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-c8ecc06c9ee4d5bb4f7af3e44ebf1554_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\lvert -49+k\\rvert}{13}=5\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"104\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E resolvemos a equa\u00e7\u00e3o resultante. No numerador da fra\u00e7\u00e3o existe um valor absoluto, portanto, devemos analisar separadamente quando o valor absoluto \u00e9 positivo e quando \u00e9 negativo: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-123\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-07410ec6d53a5693ac56545b685f08fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{+(-49+k)}{13}=5\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"130\" 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-b7e23215c12b8cd48c8dc8f53034bd57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-49+k= 5 \\cdot 13\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"125\" 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-5c7ac0f20e76f3591e7dae560e8cb7c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-49+k= 65\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"103\" 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-25f3828a5779f9ac2703a313f3032d2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k= 65+49\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" 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-29b0e2bdc51132eccc8e8e6b67301103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{k= 114}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"60\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-08ddd02fb3360a9b6a7756d13b790c95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-(-49+k)}{13}=5\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"130\" 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-368c3ad800681e7d86460e6d175d676e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"49-k= 5 \\cdot 13\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"112\" 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-aea503c1a7fc95f936b2fe2c72e946a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"49-k= 65\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"90\" 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-ef8ec1c62a0a8d710f15466eefccf911_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"49-65=k\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"90\" 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-4adfa64e85e20e7e2ada6dd8a42c5d55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{-16=k}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"63\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Existem, portanto, dois valores poss\u00edveis de<\/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> correto:<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f5172bc9d68d6bfe4b17cbe2f1e40aa0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k=114\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"60\" style=\"vertical-align: 0px;\"><\/p>\n<p> qualquer<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-796de6aa36adaf954bbb8584d094c3fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k=-16.\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"69\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Aqui voc\u00ea encontrar\u00e1 a f\u00f3rmula usada para calcular a dist\u00e2ncia entre um ponto e uma linha. Al\u00e9m disso, voc\u00ea poder\u00e1 ver diversos exemplos e exerc\u00edcios resolvidos de dist\u00e2ncias entre pontos e retas e, ainda, as aplica\u00e7\u00f5es que esta opera\u00e7\u00e3o possui (por exemplo, encontrar a dist\u00e2ncia entre retas paralelas). F\u00f3rmula para a dist\u00e2ncia entre um ponto &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/distancia-entre-um-ponto-e-uma-formula-de-linha-exemplos-de-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">Dist\u00e2ncia entre um ponto e uma reta<\/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-233","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>Dist\u00e2ncia entre um ponto e uma reta - 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\/distancia-entre-um-ponto-e-uma-formula-de-linha-exemplos-de-exercicios-resolvidos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Dist\u00e2ncia entre um ponto e uma reta - Mathority\" \/>\n<meta property=\"og:description\" content=\"Aqui voc\u00ea encontrar\u00e1 a f\u00f3rmula usada para calcular a dist\u00e2ncia entre um ponto e uma linha. Al\u00e9m disso, voc\u00ea poder\u00e1 ver diversos exemplos e exerc\u00edcios resolvidos de dist\u00e2ncias entre pontos e retas e, ainda, as aplica\u00e7\u00f5es que esta opera\u00e7\u00e3o possui (por exemplo, encontrar a dist\u00e2ncia entre retas paralelas). 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