{"id":567200,"date":"2026-07-22T15:03:51","date_gmt":"2026-07-22T08:03:51","guid":{"rendered":"https:\/\/hoanghamobile.com\/tin-tuc\/?p=567200"},"modified":"2026-07-22T15:21:13","modified_gmt":"2026-07-22T08:21:13","slug":"phuong-trinh-duong-thang","status":"publish","type":"post","link":"https:\/\/hoanghamobile.com\/tin-tuc\/phuong-trinh-duong-thang\/","title":{"rendered":"Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng: c\u00e1c d\u1ea1ng, h\u1ec7 s\u1ed1 g\u00f3c v\u00e0 v\u00ed d\u1ee5"},"content":{"rendered":"<p>Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng l\u00e0 c\u00f4ng c\u1ee5 \u0111\u1ea1i s\u1ed1 m\u00f4 t\u1ea3 ch\u00ednh x\u00e1c m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng tr\u00ean m\u1eb7t ph\u1eb3ng t\u1ecda \u0111\u1ed9, thay v\u00ec ch\u1ec9 v\u1ebd tay \u01b0\u1edbc l\u01b0\u1ee3ng. Ch\u1ec9 v\u1edbi v\u00e0i h\u1ec7 s\u1ed1, b\u1ea1n x\u00e1c \u0111\u1ecbnh \u0111\u01b0\u1ee3c \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m n\u00e0o, \u0111\u1ed9 d\u1ed1c ra sao, song song hay vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng n\u00e0o kh\u00e1c. Trong ch\u01b0\u01a1ng tr\u00ecnh To\u00e1n l\u1edbp 9 v\u00e0 l\u1edbp 10, ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 5 d\u1ea1ng vi\u1ebft ch\u00ednh: t\u1ed5ng qu\u00e1t, h\u1ec7 s\u1ed1 g\u00f3c, tham s\u1ed1, ch\u00ednh t\u1eafc v\u00e0 \u0111o\u1ea1n ch\u1eafn, m\u1ed7i d\u1ea1ng ph\u00f9 h\u1ee3p m\u1ed9t ki\u1ec3u d\u1eef ki\u1ec7n \u0111\u1ec1 b\u00e0i. N\u1ed9i dung d\u01b0\u1edbi \u0111\u00e2y t\u1ed5ng h\u1ee3p \u0111\u1ea7y \u0111\u1ee7 c\u00e1c d\u1ea1ng v\u00e0 v\u00ed d\u1ee5 c\u00f3 l\u1eddi gi\u1ea3i t\u1eebng b\u01b0\u1edbc \u0111\u1ec3 b\u1ea1n luy\u1ec7n t\u1eadp.<\/p>\n<h2>Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng l\u00e0 g\u00ec?<\/h2>\n<p>Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng l\u00e0 h\u1ec7 th\u1ee9c \u0111\u1ea1i s\u1ed1 m\u00e0 m\u1ecdi \u0111i\u1ec3m n\u1eb1m tr\u00ean \u0111\u01b0\u1eddng th\u1eb3ng \u0111\u1ec1u th\u1ecfa m\u00e3n, v\u00e0 ng\u01b0\u1ee3c l\u1ea1i. Trong m\u1eb7t ph\u1eb3ng t\u1ecda \u0111\u1ed9 Oxy, m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh khi bi\u1ebft m\u1ed9t \u0111i\u1ec3m m\u00e0 n\u00f3 \u0111i qua c\u00f9ng v\u1edbi h\u01b0\u1edbng c\u1ee7a n\u00f3, th\u1ec3 hi\u1ec7n qua h\u1ec7 s\u1ed1 g\u00f3c, vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng ho\u1eb7c vect\u01a1 ph\u00e1p tuy\u1ebfn. T\u1eeb hai d\u1eef ki\u1ec7n n\u00e0y, b\u1ea1n vi\u1ebft \u0111\u01b0\u1ee3c ph\u01b0\u01a1ng tr\u00ecnh d\u01b0\u1edbi nhi\u1ec1u d\u1ea1ng kh\u00e1c nhau v\u00e0 c\u00f3 th\u1ec3 chuy\u1ec3n \u0111\u1ed5i qua l\u1ea1i l\u1eabn nhau.<\/p>\n<p><img fetchpriority=\"high\" decoding=\"async\" class=\"size-full wp-image-568966 aligncenter\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-1.jpg\" alt=\"phuong-trinh-duong-thang-1\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-1.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-1-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-1-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>C\u00e1c d\u1ea1ng ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng th\u01b0\u1eddng g\u1eb7p<\/h2>\n<p>N\u0103m d\u1ea1ng ph\u01b0\u01a1ng tr\u00ecnh d\u01b0\u1edbi \u0111\u00e2y \u0111\u01b0\u1ee3c d\u00f9ng ph\u1ed5 bi\u1ebfn nh\u1ea5t, t\u00f9y theo \u0111\u1ec1 b\u00e0i cho \u0111i\u1ec3m, h\u1ec7 s\u1ed1 g\u00f3c hay vect\u01a1.<\/p>\n<table>\n<thead>\n<tr>\n<th>D\u1ea1ng<\/th>\n<th>C\u00f4ng th\u1ee9c<\/th>\n<th>Th\u01b0\u1eddng d\u00f9ng khi<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>T\u1ed5ng qu\u00e1t<\/td>\n<td>Ax + By + C = 0 (A, B kh\u00f4ng \u0111\u1ed3ng th\u1eddi b\u1eb1ng 0)<\/td>\n<td>Bi\u1ebft 1 \u0111i\u1ec3m v\u00e0 vect\u01a1 ph\u00e1p tuy\u1ebfn<\/td>\n<\/tr>\n<tr>\n<td>H\u1ec7 s\u1ed1 g\u00f3c<\/td>\n<td>y = kx + m (k = tan\u03b1)<\/td>\n<td>Bi\u1ebft \u0111\u1ed9 d\u1ed1c ho\u1eb7c 1 \u0111i\u1ec3m v\u00e0 h\u1ec7 s\u1ed1 g\u00f3c<\/td>\n<\/tr>\n<tr>\n<td>Tham s\u1ed1<\/td>\n<td>{x = x\u2080 + at; y = y\u2080 + bt}, t \u2208 R<\/td>\n<td>Bi\u1ebft 1 \u0111i\u1ec3m v\u00e0 vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng (a; b)<\/td>\n<\/tr>\n<tr>\n<td>Ch\u00ednh t\u1eafc<\/td>\n<td>(x \u2212 x\u2080)\/a = (y \u2212 y\u2080)\/b (a, b \u2260 0)<\/td>\n<td>Kh\u1eed tham s\u1ed1 t t\u1eeb d\u1ea1ng tham s\u1ed1<\/td>\n<\/tr>\n<tr>\n<td>\u0110o\u1ea1n ch\u1eafn<\/td>\n<td>x\/a + y\/b = 1 (a, b \u2260 0)<\/td>\n<td>Bi\u1ebft giao \u0111i\u1ec3m v\u1edbi hai tr\u1ee5c t\u1ecda \u0111\u1ed9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>D\u1ea1ng t\u1ed5ng qu\u00e1t v\u00e0 vect\u01a1 ph\u00e1p tuy\u1ebfn<\/h3>\n<p>D\u1ea1ng Ax + By + C = 0 g\u1eafn li\u1ec1n v\u1edbi vect\u01a1 ph\u00e1p tuy\u1ebfn n = (A; B) \u2014 vect\u01a1 vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng. Bi\u1ebft \u0111i\u1ec3m M(x\u2080; y\u2080) v\u00e0 vect\u01a1 ph\u00e1p tuy\u1ebfn n = (A; B), ph\u01b0\u01a1ng tr\u00ecnh t\u1ed5ng qu\u00e1t vi\u1ebft ngay \u0111\u01b0\u1ee3c: A(x \u2212 x\u2080) + B(y \u2212 y\u2080) = 0, sau \u0111\u00f3 khai tri\u1ec3n, r\u00fat g\u1ecdn. Kh\u00e1i ni\u1ec7m vect\u01a1 ph\u00e1p tuy\u1ebfn n\u00e0y \u0111\u01b0\u1ee3c m\u1edf r\u1ed9ng l\u00ean kh\u00f4ng gian ba chi\u1ec1u trong <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/phuong-trinh-mat-phang\/\">ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng<\/a>, n\u01a1i vect\u01a1 ph\u00e1p tuy\u1ebfn c\u00f3 ba t\u1ecda \u0111\u1ed9 thay v\u00ec hai.<\/p>\n<h3>D\u1ea1ng h\u1ec7 s\u1ed1 g\u00f3c<\/h3>\n<p>Trong y = kx + m, k g\u1ecdi l\u00e0 h\u1ec7 s\u1ed1 g\u00f3c, b\u1eb1ng tan\u03b1 v\u1edbi \u03b1 l\u00e0 g\u00f3c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng v\u00e0 chi\u1ec1u d\u01b0\u01a1ng tr\u1ee5c Ox; m l\u00e0 tung \u0111\u1ed9 g\u1ed1c \u2014 tung \u0111\u1ed9 giao \u0111i\u1ec3m v\u1edbi tr\u1ee5c Oy. k &gt; 0: \u0111\u01b0\u1eddng th\u1eb3ng \u0111i l\u00ean, h\u00e0m \u0111\u1ed3ng bi\u1ebfn. k &lt; 0: \u0111\u01b0\u1eddng th\u1eb3ng \u0111i xu\u1ed1ng, h\u00e0m ngh\u1ecbch bi\u1ebfn. k = 0: \u0111\u01b0\u1eddng th\u1eb3ng song song ho\u1eb7c tr\u00f9ng tr\u1ee5c ho\u00e0nh. T\u1eeb d\u1ea1ng t\u1ed5ng qu\u00e1t Ax + By + C = 0 v\u1edbi B \u2260 0, suy ra k = \u2212A\/B v\u00e0 m = \u2212C\/B.<\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-568967 aligncenter\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-2.jpg\" alt=\"phuong-trinh-duong-thang-2\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-2.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-2-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-2-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h3>D\u1ea1ng tham s\u1ed1 v\u00e0 d\u1ea1ng ch\u00ednh t\u1eafc<\/h3>\n<p>Bi\u1ebft \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m M\u2080(x\u2080; y\u2080) v\u00e0 c\u00f3 vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng u = (a; b) v\u1edbi a, b kh\u00f4ng \u0111\u1ed3ng th\u1eddi b\u1eb1ng 0, ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 l\u00e0 {x = x\u2080 + at; y = y\u2080 + bt}, t \u2208 R. Khi c\u1ea3 a v\u00e0 b kh\u00e1c 0, kh\u1eed t s\u1ebd \u0111\u01b0\u1ee3c d\u1ea1ng ch\u00ednh t\u1eafc (x \u2212 x\u2080)\/a = (y \u2212 y\u2080)\/b, g\u1ecdn h\u01a1n \u0111\u1ec3 so s\u00e1nh hai \u0111\u01b0\u1eddng th\u1eb3ng.<\/p>\n<h3>D\u1ea1ng \u0111o\u1ea1n ch\u1eafn<\/h3>\n<p>Khi \u0111\u01b0\u1eddng th\u1eb3ng c\u1eaft tr\u1ee5c ho\u00e0nh t\u1ea1i (a; 0) v\u00e0 c\u1eaft tr\u1ee5c tung t\u1ea1i (0; b) v\u1edbi a, b \u2260 0, ph\u01b0\u01a1ng tr\u00ecnh \u0111o\u1ea1n ch\u1eafn l\u00e0 x\/a + y\/b = 1 \u2014 ti\u1ec7n l\u1ee3i khi \u0111\u1ec1 b\u00e0i cho s\u1eb5n hai giao \u0111i\u1ec3m v\u1edbi hai tr\u1ee5c t\u1ecda \u0111\u1ed9.<\/p>\n<h2>C\u00e1ch vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua hai \u0111i\u1ec3m<\/h2>\n<p>Khi bi\u1ebft t\u1ecda \u0111\u1ed9 hai \u0111i\u1ec3m, b\u1ea1n t\u00ednh h\u1ec7 s\u1ed1 g\u00f3c tr\u01b0\u1edbc r\u1ed3i thay ng\u01b0\u1ee3c v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh \u0111\u1ec3 t\u00ecm tung \u0111\u1ed9 g\u1ed1c.<\/p>\n<p>V\u1edbi hai \u0111i\u1ec3m A(x\u2081; y\u2081) v\u00e0 B(x\u2082; y\u2082) c\u00f3 x\u2081 \u2260 x\u2082, h\u1ec7 s\u1ed1 g\u00f3c l\u00e0 k = (y\u2082 \u2212 y\u2081)\/(x\u2082 \u2212 x\u2081); thay t\u1ecda \u0111\u1ed9 m\u1ed9t \u0111i\u1ec3m v\u00e0o y = kx + m \u0111\u1ec3 t\u00ecm m. N\u1ebfu x\u2081 = x\u2082, \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh x = x\u2081 (kh\u00f4ng vi\u1ebft \u0111\u01b0\u1ee3c d\u1ea1ng y = kx + m).<\/p>\n<p><em>V\u00ed d\u1ee5:<\/em> Qua A(1; 2) v\u00e0 B(3; 6): k = (6 \u2212 2)\/(3 \u2212 1) = 2; thay A v\u00e0o y = kx + m: 2 = 2\u00b71 + m n\u00ean m = 0; ph\u01b0\u01a1ng tr\u00ecnh y = 2x, d\u1ea1ng t\u1ed5ng qu\u00e1t l\u00e0 2x \u2212 y = 0.<\/p>\n<p><strong>Xem th\u00eam: <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/dinh-ly-pytago\/\">\u0110\u1ecbnh l\u00fd Pytago: c\u00f4ng th\u1ee9c, \u0111\u1ecbnh l\u00fd \u0111\u1ea3o, v\u00ed d\u1ee5 \u00e1p d\u1ee5ng<\/a><\/strong><\/p>\n<h2>V\u1ecb tr\u00ed t\u01b0\u01a1ng \u0111\u1ed1i c\u1ee7a hai \u0111\u01b0\u1eddng th\u1eb3ng<\/h2>\n<p>So s\u00e1nh h\u1ec7 s\u1ed1 g\u00f3c v\u00e0 tung \u0111\u1ed9 g\u1ed1c c\u1ee7a hai \u0111\u01b0\u1eddng th\u1eb3ng cho bi\u1ebft ch\u00fang song song, tr\u00f9ng nhau, c\u1eaft nhau hay vu\u00f4ng g\u00f3c.<\/p>\n<ul>\n<li>Song song: k\u2081 = k\u2082 v\u00e0 m\u2081 \u2260 m\u2082.<\/li>\n<li>Tr\u00f9ng nhau: k\u2081 = k\u2082 v\u00e0 m\u2081 = m\u2082.<\/li>\n<li>C\u1eaft nhau: k\u2081 \u2260 k\u2082.<\/li>\n<li>Vu\u00f4ng g\u00f3c: k\u2081 \u00b7 k\u2082 = \u22121.<\/li>\n<\/ul>\n<p>V\u1edbi d\u1ea1ng t\u1ed5ng qu\u00e1t, A\u2081x + B\u2081y + C\u2081 = 0 v\u00e0 A\u2082x + B\u2082y + C\u2082 = 0 song song khi A\u2081\/A\u2082 = B\u2081\/B\u2082 \u2260 C\u2081\/C\u2082, tr\u00f9ng nhau khi c\u1ea3 ba t\u1ec9 s\u1ed1 b\u1eb1ng nhau.<\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-568968 aligncenter\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-3.jpg\" alt=\"phuong-trinh-duong-thang-3\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-3.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-3-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-3-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>V\u00ed d\u1ee5 v\u00e0 b\u00e0i t\u1eadp c\u00f3 l\u1eddi gi\u1ea3i v\u1ec1 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng<\/h2>\n<p>N\u0103m v\u00ed d\u1ee5 d\u01b0\u1edbi \u0111\u00e2y \u0111i qua t\u1eebng d\u1ea1ng ph\u01b0\u01a1ng tr\u00ecnh, k\u00e8m l\u1eddi gi\u1ea3i t\u1eebng b\u01b0\u1edbc \u0111\u1ec3 b\u1ea1n \u0111\u1ed1i chi\u1ebfu.<\/p>\n<ul>\n<li><strong>V\u00ed d\u1ee5 1 (t\u1ed5ng qu\u00e1t):<\/strong> \u0110\u01b0\u1eddng th\u1eb3ng qua M(2; \u22121), vect\u01a1 ph\u00e1p tuy\u1ebfn n = (3; 4). <em>L\u1eddi gi\u1ea3i:<\/em> A(x \u2212 x\u2080) + B(y \u2212 y\u2080) = 0 \u2192 3(x \u2212 2) + 4(y + 1) = 0 \u2192 3x + 4y \u2212 2 = 0.<\/li>\n<li><strong>V\u00ed d\u1ee5 2 (tham s\u1ed1, ch\u00ednh t\u1eafc):<\/strong> \u0110\u01b0\u1eddng th\u1eb3ng qua N(1; 3), vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng u = (2; \u22121). <em>L\u1eddi gi\u1ea3i:<\/em> Tham s\u1ed1: {x = 1 + 2t; y = 3 \u2212 t}. V\u00ec a, b \u2260 0, d\u1ea1ng ch\u00ednh t\u1eafc: (x \u2212 1)\/2 = (y \u2212 3)\/(\u22121).<\/li>\n<li><strong>V\u00ed d\u1ee5 3 (song song):<\/strong> X\u00e9t d\u2081: y = 2x + 3 v\u00e0 d\u2082: y = 2x \u2212 5. <em>L\u1eddi gi\u1ea3i:<\/em> k\u2081 = k\u2082 = 2, m\u2081 \u2260 m\u2082 n\u00ean d\u2081 song song d\u2082.<\/li>\n<li><strong>V\u00ed d\u1ee5 4 (vu\u00f4ng g\u00f3c):<\/strong> T\u00ecm h\u1ec7 s\u1ed1 g\u00f3c \u0111\u01b0\u1eddng th\u1eb3ng qua O, vu\u00f4ng g\u00f3c v\u1edbi y = (1\/3)x \u2212 2. <em>L\u1eddi gi\u1ea3i:<\/em> k\u2082 = 1\/3; k\u2081 \u00b7 k\u2082 = \u22121 n\u00ean k\u2081 = \u22123; \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm: y = \u22123x.<\/li>\n<li><strong>V\u00ed d\u1ee5 5 (\u0111o\u1ea1n ch\u1eafn):<\/strong> \u0110\u01b0\u1eddng th\u1eb3ng c\u1eaft Ox t\u1ea1i (4; 0), c\u1eaft Oy t\u1ea1i (0; \u22122). <em>L\u1eddi gi\u1ea3i:<\/em> x\/4 + y\/(\u22122) = 1 \u2192 quy \u0111\u1ed3ng, r\u00fat g\u1ecdn: x \u2212 2y \u2212 4 = 0.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-568969 aligncenter\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-4.jpg\" alt=\"phuong-trinh-duong-thang-4\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-4.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-4-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-4-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>\u1ee8ng d\u1ee5ng c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong th\u1ef1c t\u1ebf<\/h2>\n<p>Ngo\u00e0i b\u00e0i ki\u1ec3m tra, ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng c\u00f2n m\u00f4 t\u1ea3 nhi\u1ec1u m\u1ed1i quan h\u1ec7 tuy\u1ebfn t\u00ednh trong \u0111\u1eddi s\u1ed1ng v\u00e0 k\u1ef9 thu\u1eadt.<\/p>\n<p>Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng m\u00f4 t\u1ea3 quan h\u1ec7 tuy\u1ebfn t\u00ednh gi\u1eefa hai \u0111\u1ea1i l\u01b0\u1ee3ng, nh\u01b0 qu\u00e3ng \u0111\u01b0\u1eddng theo th\u1eddi gian trong chuy\u1ec3n \u0111\u1ed9ng \u0111\u1ec1u, ho\u1eb7c \u0111\u01b0\u1eddng xu h\u01b0\u1edbng khi ph\u00e2n t\u00edch s\u1ed1 li\u1ec7u. Trong l\u1eadp tr\u00ecnh \u0111\u1ed3 h\u1ecda, n\u00f3 gi\u00fap x\u00e1c \u0111\u1ecbnh v\u1ecb tr\u00ed, v\u1ebd h\u00ecnh v\u00e0 t\u00ednh giao \u0111i\u1ec3m. N\u1eafm v\u1eefng c\u00e1c d\u1ea1ng ph\u01b0\u01a1ng tr\u00ecnh c\u0169ng l\u00e0 n\u1ec1n t\u1ea3ng \u0111\u1ec3 h\u1ecdc ti\u1ebfp parabol, \u0111\u01b0\u1eddng tr\u00f2n v\u00e0 c\u00e1c \u0111\u01b0\u1eddng cong b\u1eadc cao h\u01a1n.<\/p>\n<p><strong>Xem th\u00eam: <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/dien-truong-la-gi\/\">\u0110i\u1ec7n tr\u01b0\u1eddng l\u00e0 g\u00ec? C\u01b0\u1eddng \u0111\u1ed9 \u0111i\u1ec7n tr\u01b0\u1eddng v\u00e0 c\u00f4ng th\u1ee9c E = F\/q<\/a><\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-568970 aligncenter\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-5.jpg\" alt=\"phuong-trinh-duong-thang-5\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-5.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-5-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/phuong-trinh-duong-thang-5-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>C\u00e2u h\u1ecfi th\u01b0\u1eddng g\u1eb7p v\u1ec1 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng<\/h2>\n<h3>Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 nh\u1eefng d\u1ea1ng n\u00e0o?<\/h3>\n<p>C\u00f3 5 d\u1ea1ng th\u01b0\u1eddng g\u1eb7p: t\u1ed5ng qu\u00e1t Ax + By + C = 0, h\u1ec7 s\u1ed1 g\u00f3c y = kx + m, tham s\u1ed1 {x = x\u2080 + at; y = y\u2080 + bt}, ch\u00ednh t\u1eafc (x \u2212 x\u2080)\/a = (y \u2212 y\u2080)\/b v\u00e0 \u0111o\u1ea1n ch\u1eafn x\/a + y\/b = 1.<\/p>\n<h3>H\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng l\u00e0 g\u00ec?<\/h3>\n<p>H\u1ec7 s\u1ed1 g\u00f3c l\u00e0 k trong ph\u01b0\u01a1ng tr\u00ecnh y = kx + m, b\u1eb1ng tan\u03b1 v\u1edbi \u03b1 l\u00e0 g\u00f3c h\u1ee3p b\u1edfi \u0111\u01b0\u1eddng th\u1eb3ng v\u00e0 chi\u1ec1u d\u01b0\u01a1ng tr\u1ee5c Ox; k cho bi\u1ebft \u0111\u1ed9 d\u1ed1c v\u00e0 chi\u1ec1u bi\u1ebfn thi\u00ean c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng.<\/p>\n<h3>C\u00e1ch vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua hai \u0111i\u1ec3m nh\u01b0 th\u1ebf n\u00e0o?<\/h3>\n<p>T\u00ednh h\u1ec7 s\u1ed1 g\u00f3c k = (y\u2082 \u2212 y\u2081)\/(x\u2082 \u2212 x\u2081) r\u1ed3i thay t\u1ecda \u0111\u1ed9 m\u1ed9t \u0111i\u1ec3m v\u00e0o y = kx + m \u0111\u1ec3 t\u00ecm m. N\u1ebfu hai \u0111i\u1ec3m c\u00f3 c\u00f9ng ho\u00e0nh \u0111\u1ed9, \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 d\u1ea1ng x = h\u1eb1ng s\u1ed1.<\/p>\n<h3>Ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d\u00f9ng khi n\u00e0o?<\/h3>\n<p>Khi bi\u1ebft m\u1ed9t \u0111i\u1ec3m v\u00e0 m\u1ed9t vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng (a; b), b\u1ea1n vi\u1ebft \u0111\u01b0\u1ee3c ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 {x = x\u2080 + at; y = y\u2080 + bt}, ti\u1ec7n cho c\u00e1c b\u00e0i to\u00e1n m\u00f4 t\u1ea3 chuy\u1ec3n \u0111\u1ed9ng ho\u1eb7c h\u00ecnh h\u1ecdc kh\u00f4ng gian.<\/p>\n<h3>Hai \u0111\u01b0\u1eddng th\u1eb3ng song song khi n\u00e0o?<\/h3>\n<p>Khi hai \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 c\u00f9ng h\u1ec7 s\u1ed1 g\u00f3c (k\u2081 = k\u2082) nh\u01b0ng kh\u00e1c tung \u0111\u1ed9 g\u1ed1c (m\u2081 \u2260 m\u2082). N\u1ebfu c\u1ea3 h\u1ec7 s\u1ed1 g\u00f3c v\u00e0 tung \u0111\u1ed9 g\u1ed1c \u0111\u1ec1u b\u1eb1ng nhau th\u00ec hai \u0111\u01b0\u1eddng th\u1eb3ng tr\u00f9ng nhau.<\/p>\n<h3>Hai \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c khi n\u00e0o?<\/h3>\n<p>Khi t\u00edch hai h\u1ec7 s\u1ed1 g\u00f3c b\u1eb1ng \u22121, t\u1ee9c k\u2081 \u00b7 k\u2082 = \u22121.<\/p>\n<h3>L\u00e0m sao x\u00e1c \u0111\u1ecbnh vect\u01a1 ph\u00e1p tuy\u1ebfn t\u1eeb ph\u01b0\u01a1ng tr\u00ecnh t\u1ed5ng qu\u00e1t?<\/h3>\n<p>T\u1eeb Ax + By + C = 0, vect\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 n = (A; B); vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng t\u01b0\u01a1ng \u1ee9ng vu\u00f4ng g\u00f3c v\u1edbi n l\u00e0 u = (B; \u2212A) ho\u1eb7c (\u2212B; A).<\/p>\n<p><script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[\n{\"@type\":\"Question\",\"name\":\"Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 nh\u1eefng d\u1ea1ng n\u00e0o?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"C\u00f3 5 d\u1ea1ng th\u01b0\u1eddng g\u1eb7p: t\u1ed5ng qu\u00e1t Ax + By + C = 0, h\u1ec7 s\u1ed1 g\u00f3c y = kx + m, tham s\u1ed1 x = x0 + at v\u00e0 y = y0 + bt, ch\u00ednh t\u1eafc (x-x0)\/a = (y-y0)\/b v\u00e0 \u0111o\u1ea1n ch\u1eafn x\/a + y\/b = 1.\"}},\n{\"@type\":\"Question\",\"name\":\"H\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng l\u00e0 g\u00ec?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"H\u1ec7 s\u1ed1 g\u00f3c l\u00e0 k trong ph\u01b0\u01a1ng tr\u00ecnh y = kx + m, b\u1eb1ng tan\u03b1 v\u1edbi \u03b1 l\u00e0 g\u00f3c h\u1ee3p b\u1edfi \u0111\u01b0\u1eddng th\u1eb3ng v\u00e0 chi\u1ec1u d\u01b0\u01a1ng tr\u1ee5c Ox; k cho bi\u1ebft \u0111\u1ed9 d\u1ed1c v\u00e0 chi\u1ec1u bi\u1ebfn thi\u00ean c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng.\"}},\n{\"@type\":\"Question\",\"name\":\"C\u00e1ch vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua hai \u0111i\u1ec3m nh\u01b0 th\u1ebf n\u00e0o?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"T\u00ednh h\u1ec7 s\u1ed1 g\u00f3c k = (y2 - y1)\/(x2 - x1) r\u1ed3i thay t\u1ecda \u0111\u1ed9 m\u1ed9t \u0111i\u1ec3m v\u00e0o y = kx + m \u0111\u1ec3 t\u00ecm m. N\u1ebfu hai \u0111i\u1ec3m c\u00f3 c\u00f9ng ho\u00e0nh \u0111\u1ed9, \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 d\u1ea1ng x = h\u1eb1ng s\u1ed1.\"}},\n{\"@type\":\"Question\",\"name\":\"Ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d\u00f9ng khi n\u00e0o?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Khi bi\u1ebft m\u1ed9t \u0111i\u1ec3m v\u00e0 m\u1ed9t vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng (a; b), b\u1ea1n vi\u1ebft \u0111\u01b0\u1ee3c ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 x = x0 + at v\u00e0 y = y0 + bt, ti\u1ec7n cho c\u00e1c b\u00e0i to\u00e1n m\u00f4 t\u1ea3 chuy\u1ec3n \u0111\u1ed9ng ho\u1eb7c h\u00ecnh h\u1ecdc kh\u00f4ng gian.\"}},\n{\"@type\":\"Question\",\"name\":\"Hai \u0111\u01b0\u1eddng th\u1eb3ng song song khi n\u00e0o?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Khi hai \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 c\u00f9ng h\u1ec7 s\u1ed1 g\u00f3c (k1 = k2) nh\u01b0ng kh\u00e1c tung \u0111\u1ed9 g\u1ed1c (m1 kh\u00e1c m2). N\u1ebfu c\u1ea3 h\u1ec7 s\u1ed1 g\u00f3c v\u00e0 tung \u0111\u1ed9 g\u1ed1c \u0111\u1ec1u b\u1eb1ng nhau th\u00ec hai \u0111\u01b0\u1eddng th\u1eb3ng tr\u00f9ng nhau.\"}},\n{\"@type\":\"Question\",\"name\":\"Hai \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c khi n\u00e0o?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Khi t\u00edch hai h\u1ec7 s\u1ed1 g\u00f3c b\u1eb1ng -1, t\u1ee9c k1 \u00b7 k2 = -1.\"}},\n{\"@type\":\"Question\",\"name\":\"L\u00e0m sao x\u00e1c \u0111\u1ecbnh vect\u01a1 ph\u00e1p tuy\u1ebfn t\u1eeb ph\u01b0\u01a1ng tr\u00ecnh t\u1ed5ng qu\u00e1t?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"T\u1eeb Ax + By + C = 0, vect\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 n = (A; B); vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng t\u01b0\u01a1ng \u1ee9ng vu\u00f4ng g\u00f3c v\u1edbi n l\u00e0 u = (B; -A) ho\u1eb7c (-B; A).\"}}\n]}\n<\/script><\/p>\n<h2>K\u1ebft lu\u1eadn<\/h2>\n<p>Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 5 d\u1ea1ng ch\u00ednh \u2014 t\u1ed5ng qu\u00e1t, h\u1ec7 s\u1ed1 g\u00f3c, tham s\u1ed1, ch\u00ednh t\u1eafc v\u00e0 \u0111o\u1ea1n ch\u1eafn \u2014 c\u00f3 th\u1ec3 chuy\u1ec3n \u0111\u1ed5i qua l\u1ea1i t\u00f9y d\u1eef ki\u1ec7n \u0111\u1ec1 b\u00e0i. H\u1ec7 s\u1ed1 g\u00f3c k quy\u1ebft \u0111\u1ecbnh \u0111\u1ed9 d\u1ed1c; so s\u00e1nh k v\u00e0 m gi\u1eefa hai \u0111\u01b0\u1eddng th\u1eb3ng cho bi\u1ebft ch\u00fang song song, c\u1eaft nhau hay vu\u00f4ng g\u00f3c. B\u1ea1n c\u00f3 th\u1ec3 xem th\u00eam <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/dien-tich-tam-giac\/\">c\u00f4ng th\u1ee9c t\u00ednh di\u1ec7n t\u00edch tam gi\u00e1c<\/a>, <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/dien-tich-hinh-tron\/\">c\u00f4ng th\u1ee9c t\u00ednh di\u1ec7n t\u00edch h\u00ecnh tr\u00f2n<\/a> v\u00e0 <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/cong-thuc-to-hop-chinh-hop\/\">c\u00f4ng th\u1ee9c t\u1ed5 h\u1ee3p, ch\u1ec9nh h\u1ee3p<\/a> tr\u00ean chuy\u00ean m\u1ee5c tin t\u1ee9c Ho\u00e0ng H\u00e0 Mobile.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng l\u00e0 c\u00f4ng c\u1ee5 \u0111\u1ea1i s\u1ed1 m\u00f4 t\u1ea3 ch\u00ednh x\u00e1c m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng tr\u00ean m\u1eb7t ph\u1eb3ng t\u1ecda \u0111\u1ed9, thay v\u00ec ch\u1ec9 v\u1ebd tay \u01b0\u1edbc l\u01b0\u1ee3ng. Ch\u1ec9 v\u1edbi v\u00e0i h\u1ec7 s\u1ed1, b\u1ea1n x\u00e1c \u0111\u1ecbnh \u0111\u01b0\u1ee3c \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m n\u00e0o, \u0111\u1ed9 d\u1ed1c ra sao, song song hay vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng n\u00e0o [&hellip;]<\/p>\n","protected":false},"author":65,"featured_media":568965,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng","_yoast_wpseo_title":"Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng: c\u00e1c d\u1ea1ng v\u00e0 v\u00ed d\u1ee5","_yoast_wpseo_metadesc":"Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng g\u1ed3m nh\u1eefng d\u1ea1ng n\u00e0o, h\u1ec7 s\u1ed1 g\u00f3c t\u00ednh ra sao? Xem c\u00f4ng th\u1ee9c \u0111\u1ea7y \u0111\u1ee7, v\u00ed d\u1ee5 c\u00f3 l\u1eddi gi\u1ea3i t\u1eebng b\u01b0\u1edbc v\u00e0 b\u00e0i t\u1eadp t\u1ea1i Ho\u00e0ng H\u00e0 Mobile.","footnotes":""},"categories":[16],"tags":[],"class_list":["post-567200","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-thu-thuat"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.3 (Yoast SEO v27.4) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng: c\u00e1c d\u1ea1ng v\u00e0 v\u00ed d\u1ee5<\/title>\n<meta name=\"description\" content=\"Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng g\u1ed3m nh\u1eefng d\u1ea1ng n\u00e0o, h\u1ec7 s\u1ed1 g\u00f3c t\u00ednh ra sao? 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