{"id":3646,"date":"2024-01-30T12:29:07","date_gmt":"2024-01-30T06:29:07","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3646"},"modified":"2024-10-29T15:29:15","modified_gmt":"2024-10-29T09:29:15","slug":"area-of-triangle","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/area-of-triangle\/","title":{"rendered":"\u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2, \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09ad\u09b0\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0"},"content":{"rendered":"

\u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2\u09c7\u09b0 \u09b8\u09c2\u09a4\u09cd\u09b0 (Forumla for Area of Triangle)<\/b><\/span><\/h2>\n

\u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09b6\u09c0\u09b0\u09cd\u09b7\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09cd\u09b0\u09af\u09bc\u09c7\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09a6\u09c7\u09af\u09bc\u09be \u0986\u099b\u09c7, \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u099f\u09bf\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2<\/a><\/span> \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/b><\/span><\/p>\n

\"\u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0<\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \\Delta ABC\u00a0 <\/span><\/span> \u098f\u09b0 \u09b6\u09c0\u09b0\u09cd\u09b7\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09bf \\left(x_{1}, y_{1}\\right), B\\left(x_{2}, y_{2}\\right), C\\left(x_{3}, y_{3}\\right)<\/span><\/span><\/p>\n

A, B, C \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 x – \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 AM, <\/span>\\mathrm{BL}, \\mathrm{CN}<\/span> \u09b2\u09ae\u09cd\u09ac \u0986\u0981\u0995\u09bf\u0964 \u09a4\u09be\u09b9\u09b2\u09c7, L N=O N-O L=x_{3}-x_{2}<\/span><\/span>\u00a0<\/span><\/p>\n

L M=O M-O L=x_{1}-x_{2} 9<\/span> <\/span>\u098f\u09ac\u0982 M N=O N-O M=x_{3}-x_{1}<\/span><\/span><\/p>\n

\u2234 \u2206ABC<\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2, <\/span>\u099f\u09cd\u09b0\u09be\u09aa\u09bf\u099c\u09bf\u09af\u09bc\u09be\u09ae ABLM \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 + \u099f\u09cd\u09b0\u09be\u09aa\u09bf\u099c\u09bf\u09af\u09bc\u09be\u09ae AMNC \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 – \u099f\u09cd\u09b0\u09be\u09aa\u09bf\u099c\u09bf\u09af\u09bc\u09be\u09ae BLNC \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2,<\/span><\/p>\n=\\frac{1}{2}(A M+B L) \\cdot L M+\\frac{1}{2}(A M+C N) \\cdot M N-\\frac{1}{2}(B L+C N) \\cdot L N<\/span>\n=\\frac{1}{2}\\left\\{\\left(y_{1}+y_{2}\\right) \\cdot\\left(x_{1}-x_{2}\\right)+\\left(y_{1}+y_{3}\\right)\\left(x_{3}-x_{1}\\right)-\\left(y_{2}+y_{3}\\right)\\left(x_{3}-x_{2}\\right)\\right\\}<\/span>\n

=\\frac{1}{2}\\left\\{x_{1}\\left(y_{1}+y_{2}-y_{1}-y_{3}\\right)+x_{2}\\left(y_{2}+y_{3}-y_{1}-y_{2}\\right)+x_{3}\\left(y_{1}+y_{3}-y_{2}-y_{3}\\right)\\right\\}<\/span><\/span><\/p>\n

\\therefore \\Delta A B C<\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2, =\\frac{1}{2}\\left\\{x_{1}\\left(y_{2}-y_{1}\\right)+x_{2}\\left(y_{3}-y_{1}\\right)+x_{3}\\left(y_{1}-y_{2}\\right)\\right\\}<\/span><\/span>\u00a0<\/span><\/p>\n

=\\frac{1}{2}\\left|\\begin{array}{lll}x_{1} & y_{1} & 1 \\\\ x_{2} & y_{2} & 1 \\\\ x_{3} & y_{3} & 1\\end{array}\\right|<\/span>\u00a0 [\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u09c7\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09c7]\u2026\u2026\u2026(ii)<\/p>\n=\\frac{1}{2}\\left|\\begin{array}{llll}x_{1} & -x_{2} & x_{3} & -x_{1} \\\\ y_{1} & -y_{2} & y_{3} & -y_{1}\\end{array}\\right|\\\\<\/span>\n=\\frac{1}{2}\\left\\{\\begin{array}{lll}x_{1} & y_{1} \\\\ x_{2} & y_{2}\\end{array}|+| \\begin{array}{ll}x_{2} & y_{2} \\\\ x_{3} & y_{3}\\end{array}|+| \\begin{array}{ll}x_{3} & y_{3} \\\\ x_{1} & y_{1}\\end{array} \\mid\\right\\}\\\\<\/span>\n=\\frac{1}{2}\\left|\\begin{array}{llll}x_{1} & -x_{2} & x_{3} & -x_{1} \\\\ y_{1} & -y_{2} & y_{3} & -y_{1}\\end{array}\\right|\\\\<\/span>\n

=\\frac{1}{2}\\left\\{\\left|\\begin{array}{ll}x_{1} & y_{1} \\\\ x_{2} & y_{2}\\end{array}\\right|+\\left|\\begin{array}{ll}x_{2} & y_{2} \\\\ x_{3} & y_{3}\\end{array}\\right|+\\left|\\begin{array}{ll}x_{3} & y_{3} \\\\ x_{1} & y_{1}\\end{array}\\right|\\right\\} \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots<\/span> (iii)<\/p>\n

\u09a4\u09a6\u09cd\u09b0\u09c1\u09aa, A\\left(x_{1}, y_{1}\\right), B\\left(x_{2}, y_{2}\\right), C\\left(x_{3}, y_{3}\\right), D\\left(x_{4}, y_{4}\\right)<\/span> \u09b6\u09c0\u09b0\u09cd\u09b7\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u099a\u09a4\u09c1\u09b0\u09cd\u09ad\u09c1\u099c\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2,<\/p>\n=\\frac{1}{2}\\left\\{\\left|\\begin{array}{ll}\n\nx_{1} & y_{1} \\\\\n\nx_{2} & y_{2}\n\n\\end{array}\\right|+\\left|\\begin{array}{ll}\n\nx_{2} & y_{2} \\\\\n\nx_{3} & y_{3}\n\n\\end{array}\\right|+\\left|\\begin{array}{ll}\n\nx_{3} & y_{3} \\\\\n\nx_{1} & y_{1}\n\n\\end{array}\\right|+\\left|\\begin{array}{ll}\n\nx_{4} & y_{4} \\\\\n\nx_{1} & y_{1}\n\n\\end{array}\\right|\\right\\} . . (iii)<\/span>\n

\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc\u09c7\u09b0 \u09b8\u09ae\u09af\u09bc \u09b6\u09c0\u09b0\u09cd\u09b7\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09bf \u0998\u09a1\u09bc\u09bf\u09b0 \u0995\u09be\u099f\u09be\u09b0 \u0989\u09b2\u09cd\u099f\u09be \u09a6\u09bf\u0995\u09c7 \u09ac\u09be \u0998\u09a1\u09bc\u09bf\u09b0 \u0995\u09be\u099f\u09be\u09b0 \u09a6\u09bf\u0995\u09c7 \u09a8\u09bf\u09b2\u09c7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09ac\u09be \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u09b9\u09ac\u09c7\u0964 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u09aa\u09b0\u09ae\u09ae\u09be\u09a8 \u098f\u0995\u0987 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

 <\/p>\n

\u098f\u0996\u09be\u09a8\u09c7,\u00a0 \\delta_{A B C}=\\left|\\begin{array}{lll}\n\nx_{1} & y_{1} & 1 \\\\\n\nx_{2} & y_{2} & 1 \\\\\n\nx_{3} & y_{3} & 1\n\n\\end{array}\\right| <\/span><\/span><\/p>\n =\\left|\\begin{array}{llll}\n\nx_{1} & x_{2} & x_{3} & x_{1} \\\\\n\ny_{1} & y_{2} & y_{3} & y_{1}\n\n\\end{array}\\right| <\/span>\n \\therefore \\Delta A B C=\\frac{1}{2} \\delta_{A B C} <\/span>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 =\\frac{1}{2}\\left|x_{1} y_{2}+x_{2} y_{3}+x_{3} y_{1}-y_{1} x_{2}-y_{2} x_{3}-y_{3} x_{1}\\right| <\/span><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 =\\frac{1}{2}\\left|\\left(x_{1}-x_{2}\\right)\\left(y_{2}-y_{3}\\right)-\\left(y_{1}-y_{2}\\right)\\left(x_{2}-x_{3}\\right)\\right| <\/span><\/span><\/p>\n

\\left(x_{1}, y_{1}\\right)\\left(x_{2}, y_{2}\\right)\\left(x_{3}, y_{3}\\right) \\ldots \\ldots \\ldots \\ldots\\left(x_{n}, y_{n}\\right) <\/span><\/span> \u09b6\u09c0\u09b0\u09cd\u09b7 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09a0\u09bf\u09a4 \u09ac\u09b9\u09c1\u09ad\u09c1\u099c\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2,<\/span><\/p>\n =\\frac{1}{2}\\left|\\begin{array}{lllllll}\n\nx_{1} & x_{2} & x_{3} & \\ldots & \\ldots & x_{n} & x_{1} \\\\\n\ny_{1} & y_{2} & y_{3} & \\cdots & \\cdots & \\cdots & y_{n} & y_{1}\n\n\\end{array}\\right| <\/span>\n =\\frac{1}{2}\\left|x_{1} y_{2}+x_{2} y_{3}+\\cdots x n y_{1}-y_{1} x_{2}-y_{2} x_{3}-\\cdots y n x_{1}\\right| <\/span>\n