{"id":2095,"date":"2026-10-14T19:36:35","date_gmt":"2026-10-14T18:36:35","guid":{"rendered":"https:\/\/wpcalc.com\/pt\/matematica\/distancia-ponto-reta\/"},"modified":"2026-10-14T19:36:35","modified_gmt":"2026-10-14T18:36:35","slug":"distancia-ponto-reta","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pt\/matematica\/distancia-ponto-reta\/","title":{"rendered":"Calculadora de dist\u00e2ncia de ponto a reta"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Encontre a dist\u00e2ncia perpendicular de um ponto a uma reta<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"2095\" data-favorite-title=\"Dist\u00e2ncia de ponto a reta\" data-favorite-url=\"https:\/\/wpcalc.com\/pt\/matematica\/distancia-ponto-reta\/\" data-favorite-excerpt=\"Digite as coordenadas de um ponto e os coeficientes de uma reta na forma Ax + By + C = 0 para encontrar a menor dist\u00e2ncia (perpendicular) entre eles, com todas as etapas do c\u00e1lculo.\" aria-label=\"Adicionar aos favoritos\" data-favorite-icon=\"icon icon-geometry\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/pt\/matematica\/distancia-ponto-reta\/\" method=\"POST\" class=\"calculator\" id=\"calculator-2095\" data-post=\"2095\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Campos de entrada<\/legend><div class=\"is-single field\" id=\"input-1\"><span class=\"font-semibold\">Ponto P<\/span><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Coordenada x (x\u2081)<\/label><div class=\"term\">x\u2081<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"coordenada_x_x\" id=\"field-2\" aria-describedby=\"calc-info-2\" placeholder=\"3\" step=\"any\" value=\"3\"\/><small id=\"calc-info-2\">Coordenada horizontal do ponto.<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Coordenada y (y\u2081)<\/label><div class=\"term\">y\u2081<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"coordenada_y_y\" id=\"field-3\" aria-describedby=\"calc-info-3\" placeholder=\"4\" step=\"any\" value=\"4\"\/><small id=\"calc-info-3\">Coordenada vertical do ponto.<\/small><\/div><div class=\"is-single field\" id=\"input-4\"><span class=\"font-semibold\">Reta Ax + By + C = 0<\/span><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-5\">Coeficiente A<\/label><div class=\"term\">A<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"coeficiente_a\" id=\"field-5\" aria-describedby=\"calc-info-5\" placeholder=\"2\" step=\"any\" value=\"2\"\/><small id=\"calc-info-5\">Coeficiente de x na equa\u00e7\u00e3o da reta.<\/small><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">Coeficiente B<\/label><div class=\"term\">B<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"coeficiente_b\" id=\"field-6\" aria-describedby=\"calc-info-6\" placeholder=\"-1\" step=\"any\" value=\"-1\"\/><small id=\"calc-info-6\">Coeficiente de y na equa\u00e7\u00e3o da reta.<\/small><\/div><div class=\"field has-term\" id=\"input-7\"><label for=\"field-7\">Coeficiente C<\/label><div class=\"term\">C<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"coeficiente_c\" id=\"field-7\" aria-describedby=\"calc-info-7\" placeholder=\"1\" step=\"any\" value=\"1\"\/><small id=\"calc-info-7\">Termo constante da equa\u00e7\u00e3o da reta.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Recalcular\" id=\"calculate-button\" data-post=\"2095\">Calcular<\/button><button type=\"reset\">Redefinir<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calcular automaticamente<\/label><small>Se ativado, o resultado \u00e9 atualizado automaticamente quando voc\u00ea altera qualquer valor.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Sua dist\u00e2ncia<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Menor dist\u00e2ncia de P at\u00e9 a reta<\/span><span class=\"result-term\">d<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Dist\u00e2ncia ao quadrado<\/span><span class=\"result-term\">d\u00b2<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-3\"><span class=\"field-title\"><span>Etapas do c\u00e1lculo<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copiar link para este c\u00e1lculo<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Deixar um coment\u00e1rio<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"2095\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">O que voc\u00ea achou deste resultado?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Curti <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">\u00datil <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confuso <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Decepcionado <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Impreciso <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>F\u00f3rmula da dist\u00e2ncia de ponto a reta<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">F\u00f3rmula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ d = \\frac{|A x_1 + B y_1 + C|}{\\sqrt{A^2 + B^2}} \\]<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Onde:<\/strong><\/p>\n<ul>\n<li>\\(d\\) \u2014 dist\u00e2ncia perpendicular do ponto \u00e0 reta (unidades)<\/li>\n<li>\\(x_1, y_1\\) \u2014 coordenadas do ponto P<\/li>\n<li>\\(A, B, C\\) \u2014 coeficientes da reta \\(Ax + By + C = 0\\)<\/li>\n<\/ul>\n<p>A f\u00f3rmula mede o quanto o ponto est\u00e1 afastado da reta pelo caminho mais curto poss\u00edvel: a perpendicular. O numerador avalia a equa\u00e7\u00e3o da reta no ponto, e o denominador \\(\\sqrt{A^2 + B^2}\\) normaliza os coeficientes para que o resultado seja uma dist\u00e2ncia euclidiana de verdade.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Dist\u00e2ncia de ponto a reta \u2013 exemplo de c\u00e1lculo<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemplo:<\/strong> encontrar a dist\u00e2ncia do ponto P(3, 4) \u00e0 reta \\(2x - y + 1 = 0\\).<\/p>\n<p>\\[ A x_1 + B y_1 + C = 2 \\cdot 3 + (-1) \\cdot 4 + 1 = 6 - 4 + 1 = 3 \\]<\/p>\n<p>\\[ d = \\frac{|3|}{\\sqrt{2^2 + (-1)^2}} = \\frac{3}{\\sqrt{5}} \\approx 1.342 \\text{ unidades} \\]<\/p>\n<p><strong>Resultado:<\/strong> a menor dist\u00e2ncia de P(3, 4) \u00e0 reta \u00e9 \\(\\frac{3}{\\sqrt{5}} \\approx 1.342\\) unidades (exatamente \\(\\frac{3\\sqrt{5}}{5}\\)), e \\(d^2 = \\frac{9}{5} = 1.8\\) unidades\u00b2.<\/p>\n<p>\n\r\n<\/section>\n<h2>Como usar a calculadora<\/h2>\n<ol>\n<li>Digite a <strong>coordenada x (x\u2081)<\/strong> do seu ponto P.<\/li>\n<li>Digite a <strong>coordenada y (y\u2081)<\/strong> do seu ponto P.<\/li>\n<li>Digite o <strong>Coeficiente A<\/strong> \u2014 o n\u00famero que multiplica \\(x\\) na equa\u00e7\u00e3o da sua reta.<\/li>\n<li>Digite o <strong>Coeficiente B<\/strong> \u2014 o n\u00famero que multiplica \\(y\\).<\/li>\n<li>Digite o <strong>Coeficiente C<\/strong> \u2014 o termo constante.<\/li>\n<li>Leia o resultado: a menor dist\u00e2ncia \\(d\\), a dist\u00e2ncia ao quadrado \\(d^2\\) e o c\u00e1lculo detalhado passo a passo.<\/li>\n<\/ol>\n<p>Se a sua reta estiver na forma reduzida \\(y = mx + b\\), reescreva-a primeiro: \\(mx - y + b = 0\\), de modo que \\(A = m\\), \\(B = -1\\) e \\(C = b\\). Para uma reta que passa por dois pontos, primeiro coloque-a na forma \\(Ax + By + C = 0\\) antes de digitar os coeficientes.<\/p>\n<h2>O que \u00e9 a dist\u00e2ncia de um ponto a uma reta?<\/h2>\n<p>Dado um ponto P e uma reta infinita, existem infinitos segmentos ligando P a pontos da reta. O mais curto deles \u00e9 o segmento perpendicular \u00e0 reta, e seu comprimento \u00e9 o que chamamos de dist\u00e2ncia do ponto \u00e0 reta. Esse \u00e9 um conceito fundamental da geometria anal\u00edtica, com aplica\u00e7\u00f5es em computa\u00e7\u00e3o gr\u00e1fica, rob\u00f3tica, detec\u00e7\u00e3o de colis\u00f5es, sistemas de informa\u00e7\u00e3o geogr\u00e1fica e otimiza\u00e7\u00e3o.<\/p>\n<p>A f\u00f3rmula funciona para qualquer reta escrita na forma geral \\(Ax + By + C = 0\\). A express\u00e3o \\(Ax_1 + By_1 + C\\) \u00e9 zero exatamente quando o ponto est\u00e1 sobre a reta. Quando n\u00e3o \u00e9 zero, a divis\u00e3o por \\(\\sqrt{A^2 + B^2}\\) \u2014 o comprimento do vetor normal \\((A, B)\\) \u2014 converte o valor bruto em uma dist\u00e2ncia real, tornando o resultado independente da escala da equa\u00e7\u00e3o. Multiplicar toda a equa\u00e7\u00e3o por 2, por exemplo, n\u00e3o altera a dist\u00e2ncia.<\/p>\n<p>Um resultado de <strong>0 unidades<\/strong> significa que o ponto est\u00e1 exatamente sobre a reta. Valores maiores indicam apenas que o ponto est\u00e1 mais afastado; a dist\u00e2ncia \u00e9 sempre n\u00e3o negativa por causa do valor absoluto no numerador. O segmento perpendicular tamb\u00e9m mostra onde est\u00e1 o ponto mais pr\u00f3ximo da reta: \u00e9 o p\u00e9 da perpendicular tra\u00e7ada a partir de P.<\/p>\n<h2>Suposi\u00e7\u00f5es, limita\u00e7\u00f5es e dicas<\/h2>\n<ul>\n<li><strong>A reta precisa ser v\u00e1lida:<\/strong> os coeficientes A e B n\u00e3o podem ser ambos zero; caso contr\u00e1rio, a equa\u00e7\u00e3o n\u00e3o define uma reta e a calculadora exibir\u00e1 um erro.<\/li>\n<li><strong>A reta \u00e9 infinita:<\/strong> a dist\u00e2ncia \u00e9 medida at\u00e9 a reta completa, e n\u00e3o at\u00e9 um segmento ou uma semirreta. No caso de um segmento, o ponto mais pr\u00f3ximo pode ser uma das extremidades.<\/li>\n<li><strong>Invari\u00e2ncia de escala:<\/strong> voc\u00ea pode multiplicar A, B e C por qualquer constante n\u00e3o nula (inclusive negativa) sem alterar a dist\u00e2ncia, pois a f\u00f3rmula normaliza automaticamente.<\/li>\n<li><strong>Convers\u00e3o de outras formas:<\/strong> de \\(y = mx + b\\), use \\(A = m\\), \\(B = -1\\), \\(C = b\\); de dois pontos \\((x_1, y_1)\\), \\((x_2, y_2)\\), use \\(A = y_1 - y_2\\), \\(B = x_2 - x_1\\), \\(C = x_1 y_2 - x_2 y_1\\).<\/li>\n<li><strong>Unidades:<\/strong> a dist\u00e2ncia \u00e9 expressa nas mesmas unidades das suas coordenadas, ent\u00e3o use unidades coerentes nos dois eixos.<\/li>\n<\/ul>\n<p>Dist\u00e2ncias entre pontos e retas aparecem em muitos problemas relacionados. Se voc\u00ea est\u00e1 reescrevendo equa\u00e7\u00f5es de retas ou resolvendo sistemas que envolvem retas perpendiculares, a <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/metodo-adicao\/\">calculadora do m\u00e9todo da elimina\u00e7\u00e3o<\/a> pode ajudar a encontrar pontos de interse\u00e7\u00e3o, enquanto expandir express\u00f5es como os produtos \\((A x_1 + B y_1 + C)\\) \u00e9 r\u00e1pido com a <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/metodo-foil\/\">calculadora do m\u00e9todo FOIL<\/a> ou a <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/multiplicar-binomios-2\/\">calculadora de multiplica\u00e7\u00e3o de bin\u00f4mios<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Digite as coordenadas de um ponto e os coeficientes de uma reta na forma Ax + By + C = 0 para encontrar a menor dist\u00e2ncia (perpendicular) entre eles, com todas as etapas do c\u00e1lculo.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[42],"class_list":["post-2095","post","type-post","status-publish","format-standard","hentry","category-matematica","tag-geometria"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/2095","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/comments?post=2095"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/2095\/revisions"}],"predecessor-version":[{"id":2747,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/2095\/revisions\/2747"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/media?parent=2095"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/categories?post=2095"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/tags?post=2095"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}