{"id":2568,"date":"2026-10-14T19:37:41","date_gmt":"2026-10-14T18:37:41","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/matematyka\/odleglosc-punktu-od-prostej\/"},"modified":"2026-10-14T19:37:41","modified_gmt":"2026-10-14T18:37:41","slug":"odleglosc-punktu-od-prostej","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/matematyka\/odleglosc-punktu-od-prostej\/","title":{"rendered":"Kalkulator odleg\u0142o\u015bci punktu od prostej"},"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\">Oblicz odleg\u0142o\u015b\u0107 punktu od prostej<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"2568\" data-favorite-title=\"Odleg\u0142o\u015b\u0107 punktu od prostej\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/matematyka\/odleglosc-punktu-od-prostej\/\" data-favorite-excerpt=\"Podaj wsp\u00f3\u0142rz\u0119dne punktu i wsp\u00f3\u0142czynniki prostej w postaci Ax + By + C = 0, aby obliczy\u0107 najkr\u00f3tsz\u0105 (prostopad\u0142\u0105) odleg\u0142o\u015b\u0107 mi\u0119dzy nimi \u2014 wraz z pe\u0142nymi krokami oblicze\u0144.\" aria-label=\"Dodaj do ulubionych\" 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\/pl\/matematyka\/odleglosc-punktu-od-prostej\/\" method=\"POST\" class=\"calculator\" id=\"calculator-2568\" data-post=\"2568\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Pola danych<\/legend><div class=\"is-single field\" id=\"input-1\"><span class=\"font-semibold\">Punkt P<\/span><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Wsp\u00f3\u0142rz\u0119dna 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=\"wsprzdna_x_x\" id=\"field-2\" aria-describedby=\"calc-info-2\" placeholder=\"3\" step=\"any\" value=\"3\"\/><small id=\"calc-info-2\">Pozioma wsp\u00f3\u0142rz\u0119dna punktu.<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Wsp\u00f3\u0142rz\u0119dna 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=\"wsprzdna_y_y\" id=\"field-3\" aria-describedby=\"calc-info-3\" placeholder=\"4\" step=\"any\" value=\"4\"\/><small id=\"calc-info-3\">Pionowa wsp\u00f3\u0142rz\u0119dna punktu.<\/small><\/div><div class=\"is-single field\" id=\"input-4\"><span class=\"font-semibold\">Prosta Ax + By + C = 0<\/span><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-5\">Wsp\u00f3\u0142czynnik 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=\"wspczynnik_a\" id=\"field-5\" aria-describedby=\"calc-info-5\" placeholder=\"2\" step=\"any\" value=\"2\"\/><small id=\"calc-info-5\">Wsp\u00f3\u0142czynnik przy x w r\u00f3wnaniu prostej.<\/small><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">Wsp\u00f3\u0142czynnik 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=\"wspczynnik_b\" id=\"field-6\" aria-describedby=\"calc-info-6\" placeholder=\"-1\" step=\"any\" value=\"-1\"\/><small id=\"calc-info-6\">Wsp\u00f3\u0142czynnik przy y w r\u00f3wnaniu prostej.<\/small><\/div><div class=\"field has-term\" id=\"input-7\"><label for=\"field-7\">Wsp\u00f3\u0142czynnik 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=\"wspczynnik_c\" id=\"field-7\" aria-describedby=\"calc-info-7\" placeholder=\"1\" step=\"any\" value=\"1\"\/><small id=\"calc-info-7\">Wyraz wolny r\u00f3wnania prostej.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"2568\">Oblicz<\/button><button type=\"reset\">Wyczy\u015b\u0107<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Obliczaj automatycznie<\/label><small>Po w\u0142\u0105czeniu wynik aktualizuje si\u0119 automatycznie po zmianie dowolnej warto\u015bci.<\/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> Twoja odleg\u0142o\u015b\u0107<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Najkr\u00f3tsza odleg\u0142o\u015b\u0107 od punktu P do prostej<\/span><span class=\"result-term\">d<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>Kwadrat odleg\u0142o\u015bci<\/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=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>Kroki oblicze\u0144<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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\">Kopiuj link do tego obliczenia<\/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>Zostaw komentarz<\/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=\"2568\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Jak oceniasz ten wynik?<\/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\">Lubi\u0119 to <\/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\">Pomocne <\/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\">Niejasne <\/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\">Rozczarowanie <\/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\">Niedok\u0142adne <\/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>Wz\u00f3r na odleg\u0142o\u015b\u0107 punktu od prostej<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Wz\u00f3r<\/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>Gdzie:<\/strong><\/p>\n<ul>\n<li>\\(d\\) \u2014 odleg\u0142o\u015b\u0107 prostopad\u0142a od punktu do prostej (w jednostkach)<\/li>\n<li>\\(x_1, y_1\\) \u2014 wsp\u00f3\u0142rz\u0119dne punktu P<\/li>\n<li>\\(A, B, C\\) \u2014 wsp\u00f3\u0142czynniki prostej \\(Ax + By + C = 0\\)<\/li>\n<\/ul>\n<p>Wz\u00f3r wyznacza, jak daleko punkt le\u017cy od prostej wzd\u0142u\u017c najkr\u00f3tszej mo\u017cliwej drogi \u2014 czyli prostopadle. Licznik to warto\u015b\u0107 r\u00f3wnania prostej w danym punkcie, a mianownik \\(\\sqrt{A^2 + B^2}\\) normalizuje wsp\u00f3\u0142czynniki, tak aby wynik by\u0142 prawdziw\u0105 odleg\u0142o\u015bci\u0105 euklidesow\u0105.<\/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>Odleg\u0142o\u015b\u0107 punktu od prostej \u2014 przyk\u0142ad obliczenia<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> oblicz odleg\u0142o\u015b\u0107 punktu P(3, 4) od prostej \\(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{ jednostki} \\]<\/p>\n<p><strong>Wynik:<\/strong> najkr\u00f3tsza odleg\u0142o\u015b\u0107 od punktu P(3, 4) do prostej wynosi \\(\\frac{3}{\\sqrt{5}} \\approx 1.342\\) jednostki (dok\u0142adnie \\(\\frac{3\\sqrt{5}}{5}\\)), a \\(d^2 = \\frac{9}{5} = 1.8\\) jednostek\u00b2.<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Wpisz <strong>wsp\u00f3\u0142rz\u0119dn\u0105 x (x\u2081)<\/strong> swojego punktu P.<\/li>\n<li>Wpisz <strong>wsp\u00f3\u0142rz\u0119dn\u0105 y (y\u2081)<\/strong> swojego punktu P.<\/li>\n<li>Wpisz <strong>wsp\u00f3\u0142czynnik A<\/strong> \u2014 liczb\u0119 stoj\u0105c\u0105 przy \\(x\\) w r\u00f3wnaniu prostej.<\/li>\n<li>Wpisz <strong>wsp\u00f3\u0142czynnik B<\/strong> \u2014 liczb\u0119 stoj\u0105c\u0105 przy \\(y\\).<\/li>\n<li>Wpisz <strong>wsp\u00f3\u0142czynnik C<\/strong> \u2014 wyraz wolny.<\/li>\n<li>Odczytaj wynik: najkr\u00f3tsz\u0105 odleg\u0142o\u015b\u0107 \\(d\\), jej kwadrat \\(d^2\\) oraz rozwi\u0105zanie krok po kroku.<\/li>\n<\/ol>\n<p>Je\u015bli prosta jest zadana w postaci kierunkowej \\(y = mx + b\\), najpierw przekszta\u0142\u0107 j\u0105 do postaci \\(mx - y + b = 0\\); wtedy \\(A = m\\), \\(B = -1\\) oraz \\(C = b\\). Je\u015bli prosta przechodzi przez dwa punkty, najpierw sprowad\u017a jej r\u00f3wnanie do postaci \\(Ax + By + C = 0\\), a dopiero potem wpisuj wsp\u00f3\u0142czynniki.<\/p>\n<h2>Czym jest odleg\u0142o\u015b\u0107 punktu od prostej?<\/h2>\n<p>Maj\u0105c punkt P i niesko\u0144czenie d\u0142ug\u0105 prost\u0105, mo\u017cemy poprowadzi\u0107 niesko\u0144czenie wiele odcink\u00f3w \u0142\u0105cz\u0105cych P z punktami prostej. Najkr\u00f3tszy z nich jest prostopad\u0142y do prostej i to w\u0142a\u015bnie jego d\u0142ugo\u015b\u0107 nazywamy odleg\u0142o\u015bci\u0105 punktu od prostej. To podstawowe poj\u0119cie geometrii analitycznej, znajduj\u0105ce zastosowanie m.in. w grafice komputerowej, robotyce, wykrywaniu kolizji, systemach informacji geograficznej czy optymalizacji.<\/p>\n<p>Wz\u00f3r dzia\u0142a dla dowolnej prostej zapisanej w postaci og\u00f3lnej \\(Ax + By + C = 0\\). Wyra\u017cenie \\(Ax_1 + By_1 + C\\) jest r\u00f3wne zeru dok\u0142adnie wtedy, gdy punkt le\u017cy na prostej. Gdy nie jest zerowe, dzielenie przez \\(\\sqrt{A^2 + B^2}\\) \u2014 czyli d\u0142ugo\u015b\u0107 wektora normalnego \\((A, B)\\) \u2014 zamienia surow\u0105 warto\u015b\u0107 na faktyczn\u0105 odleg\u0142o\u015b\u0107, dzi\u0119ki czemu wynik nie zale\u017cy od przeskalowania r\u00f3wnania. Pomno\u017cenie ca\u0142ego r\u00f3wnania np. przez 2 nie zmienia odleg\u0142o\u015bci.<\/p>\n<p>Wynik <strong>0 jednostek<\/strong> oznacza, \u017ce punkt le\u017cy dok\u0142adnie na prostej. Wi\u0119ksze warto\u015bci oznaczaj\u0105 po prostu, \u017ce punkt jest dalej; odleg\u0142o\u015b\u0107 jest zawsze nieuj\u0105ca, bo licznik staje w warto\u015bci bezwzgl\u0119dnej. Odcinek prostopad\u0142y wskazuje te\u017c, gdzie znajduje si\u0119 najbli\u017cszy punkt prostej \u2014 jest to spodek prostopad\u0142ej opuszczonej z punktu P.<\/p>\n<h2>Za\u0142o\u017cenia, ograniczenia i wskaz\u00f3wki<\/h2>\n<ul>\n<li><strong>Prosta musi by\u0107 poprawna:<\/strong> wsp\u00f3\u0142czynniki A i B nie mog\u0105 by\u0107 jednocze\u015bnie zerami; w przeciwnym razie r\u00f3wnanie nie opisuje prostej i kalkulator wy\u015bwietli b\u0142\u0105d.<\/li>\n<li><strong>Prosta jest niesko\u0144czona:<\/strong> odleg\u0142o\u015b\u0107 mierzymy do ca\u0142ej prostej, a nie do odcinka czy p\u00f3\u0142prostej. Dla odcinka najbli\u017cszym punktem mo\u017ce okaza\u0107 si\u0119 jeden z jego ko\u0144c\u00f3w.<\/li>\n<li><strong>Niezale\u017cno\u015b\u0107 od skali:<\/strong> mo\u017cesz pomno\u017cy\u0107 A, B i C przez dowoln\u0105 niezerow\u0105 sta\u0142\u0105 (tak\u017ce ujemn\u0105) \u2014 odleg\u0142o\u015b\u0107 si\u0119 nie zmieni, poniewa\u017c wz\u00f3r normalizuje wsp\u00f3\u0142czynniki automatycznie.<\/li>\n<li><strong>Zamiana z innych postaci:<\/strong> z postaci \\(y = mx + b\\) przyjmij \\(A = m\\), \\(B = -1\\), \\(C = b\\); z dw\u00f3ch punkt\u00f3w \\((x_1, y_1)\\), \\((x_2, y_2)\\) u\u017cyj \\(A = y_1 - y_2\\), \\(B = x_2 - x_1\\), \\(C = x_1 y_2 - x_2 y_1\\).<\/li>\n<li><strong>Jednostki:<\/strong> odleg\u0142o\u015b\u0107 wyra\u017cona jest w tych samych jednostkach co wsp\u00f3\u0142rz\u0119dne, wi\u0119c dbaj o sp\u00f3jno\u015b\u0107 jednostek na obu osiach.<\/li>\n<\/ul>\n<p>Odleg\u0142o\u015bci mi\u0119dzy punktami i prostymi pojawiaj\u0105 si\u0119 w wielu pokrewnych zadaniach. Je\u015bli przekszta\u0142casz r\u00f3wnania prostych albo rozwi\u0105zujesz uk\u0142ady z prostymi prostopad\u0142ymi, punkty przeci\u0119cia pomo\u017ce znale\u017a\u0107 <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/metoda-eliminacji\/\">kalkulator metody eliminacji<\/a>, a rozwijanie wyra\u017ce\u0144 w stylu iloczyn\u00f3w \\((A x_1 + B y_1 + C)\\) szybko wykonasz <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/metoda-foil\/\">kalkulatorem metody FOIL<\/a> lub <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/mnozenie-dwumianow-2\/\">kalkulatorem mno\u017cenia dwumian\u00f3w<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Podaj wsp\u00f3\u0142rz\u0119dne punktu i wsp\u00f3\u0142czynniki prostej w postaci Ax + By + C = 0, aby obliczy\u0107 najkr\u00f3tsz\u0105 (prostopad\u0142\u0105) odleg\u0142o\u015b\u0107 mi\u0119dzy nimi \u2014 wraz z pe\u0142nymi krokami oblicze\u0144.<\/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":[6],"tags":[41],"class_list":["post-2568","post","type-post","status-publish","format-standard","hentry","category-matematyka","tag-geometria"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2568","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/comments?post=2568"}],"version-history":[{"count":0,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2568\/revisions"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=2568"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=2568"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=2568"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}