{"id":21923,"date":"2024-01-08T12:37:59","date_gmt":"2024-01-08T07:07:59","guid":{"rendered":"https:\/\/apboardsolutions.in\/?p=21923"},"modified":"2024-01-09T12:01:52","modified_gmt":"2024-01-09T06:31:52","slug":"inter-1st-year-maths-1a-properties-of-triangles-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.in\/inter-1st-year-maths-1a-properties-of-triangles-formulas\/","title":{"rendered":"Inter 1st Year Maths 1A Properties of Triangles Formulas"},"content":{"rendered":"<p>Use these <a href=\"https:\/\/apboardsolutions.in\/inter-1st-year-maths-1a-formulas\/\">Inter 1st Year Maths 1A Formulas<\/a> PDF Chapter 10 Properties of Triangles to solve questions creatively.<\/p>\n<h2>Intermediate 1st Year Maths 1A Properties of Triangles Formulas<\/h2>\n<p>\u2192 Sine Rule :<br \/>\nIn \u0394ABC \\(\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}\\) = 2R where R is the circumradius of \u0394ABC.<\/p>\n<p>\u2192 Cosine Rule :<br \/>\na<sup>2<\/sup> = b<sup>2<\/sup> + c<sup>2<\/sup> &#8211; 2bc. cos A ;<br \/>\nb<sup>2<\/sup> = c<sup>2<\/sup> + a<sup>2<\/sup> &#8211; 2ca.cos B;<br \/>\nc<sup>2<\/sup> = a<sup>2<\/sup> + b<sup>2<\/sup> &#8211; 2ab. cos C.<\/p>\n<p>\u2192 cos A = \\(\\frac{b^{2}+c^{2}-a^{2}}{2 b c}\\),<br \/>\ncos B = \\(\\frac{c^{2}+a^{2}-b^{2}}{2 c a}\\),<br \/>\ncos C = \\(\\frac{a^{2}+b^{2}-c^{2}}{2 a b}\\)<\/p>\n<p>\u2192 a = b cos C + c cos B,b = c cos A + a cos C and c = a cos B + b cos A (Projection rule)<\/p>\n<p>\u2192 tan \\(\\frac{B-C}{2}=\\frac{b-c}{b+c}\\) cot\\(\\frac{A}{2}\\) (Napier&#8217;s analogy or tangent rule)<\/p>\n<ul>\n<li>sin\\(\\frac{A}{2}\\) = \\(\\sqrt{\\frac{(s-b)(s-c)}{b c}}\\)<\/li>\n<li>cos\\(\\frac{A}{2}\\) = \\(\\sqrt{\\frac{s(s-a)}{b c}}\\)<\/li>\n<li>tan\\(\\frac{A}{2}\\) = \\(\\sqrt{\\frac{(s-b)(s-c)}{s(s-a)}}=\\frac{\\Delta}{s(s-a)}\\)<\/li>\n<\/ul>\n<p>\u2192 \u0394 = area of \u0394ABC = \\(\\frac{1}{2}\\) bc sin A = \\(\\frac{1}{2}\\) ca sin B = \\(\\frac{1}{2}\\) ab sin C<br \/>\n= \\(\\sqrt{s(s-a)(s-b)(s-c)}=\\frac{a b c}{4 R}\\)<br \/>\n= 2R<sup>2<\/sup> sin A sin B sin C<\/p>\n<ul>\n<li>r = \\(\\frac{\\Delta}{s}\\)<\/li>\n<li>r<sub>1<\/sub> = \\(\\frac{\\Delta}{s-a}\\)<\/li>\n<li>r<sub>2<\/sub> = \\(\\frac{\\Delta}{s-b}\\)<\/li>\n<li>r<sub>3<\/sub> = \\(\\frac{\\Delta}{s-c}\\)<\/li>\n<\/ul>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 1st Year Maths 1A Properties of Triangles Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 r = 4 R sin \\(\\frac{A}{2}\\) sin \\(\\frac{B}{2}\\) sin \\(\\frac{C}{2}\\); r<sub>1<\/sub> = 4Rsin \\(\\frac{A}{2}\\) cos \\(\\frac{B}{2}\\) cos \\(\\frac{C}{2}\\)<\/p>\n<p>\u2192 r = (s &#8211; a) tan \\(\\frac{A}{2}\\);<br \/>\nr<sub>1<\/sub> = s tan \\(\\frac{A}{2}\\) = (s &#8211; c) cot \\(\\frac{B}{2}\\) = (s &#8211; b) cot \\(\\frac{C}{2}\\)<\/p>\n<p><strong>Mollweide rule:<\/strong><br \/>\nIn \u0394ABC \\(\\frac{a+b}{c}=\\frac{\\cos \\left(\\frac{A-B}{2}\\right)}{\\sin \\frac{C}{2}}\\)<br \/>\n\\(\\frac{b+c}{a}=\\frac{\\cos \\left(\\frac{B-C}{2}\\right)}{\\sin \\frac{A}{2}}\\)<br \/>\n\\(\\frac{c+a}{b}=\\frac{\\cos \\left(\\frac{C-A}{2}\\right)}{\\sin \\frac{B}{2}}\\)<\/p>\n<p><strong>Sine rule :<\/strong><br \/>\nIn \u0394ABC, \\(\\frac{\\mathrm{a}}{\\sin \\mathrm{A}}=\\frac{\\mathrm{b}}{\\sin \\mathrm{B}}=\\frac{\\mathrm{c}}{\\sin \\mathrm{C}}\\) = 2R Where R is the circum &#8211; radius.<br \/>\n\u21d2 a = 2R sin A, b = 2R sin B, c = 2R sin C<br \/>\na : b : c = sin A : sin B : sin C.<\/p>\n<p><strong>Cosine rule :<\/strong><br \/>\nIn \u0394ABC,<br \/>\na<sup>2<\/sup> = b<sup>2<\/sup> + c<sup>2<\/sup> &#8211; 2bc cos A<br \/>\nb<sup>2<\/sup> = c<sup>2<\/sup> + a<sup>2<\/sup> &#8211; 2ac cos B<br \/>\nc<sup>2<\/sup> = a<sup>2<\/sup> + b<sup>2<\/sup> &#8211; 2ab cos C<br \/>\nor<br \/>\ncos A = \\(\\frac{\\mathrm{b}^{2}+\\mathrm{c}^{2}-\\mathrm{a}^{2}}{2 \\mathrm{bc}}\\)<br \/>\ncos B = \\(\\frac{c^{2}+a^{2}-b^{2}}{2 c a}\\)<br \/>\ncos C = \\(\\frac{a^{2}+b^{2}-c^{2}}{2 a b}\\) \u21d2 cos A : cos B : cos C<br \/>\n= a(b<sup>2<\/sup> + c<sup>2<\/sup> &#8211; a<sup>2<\/sup>) : b(c<sup>2<\/sup> + a<sup>2<\/sup> &#8211; b<sup>2<\/sup>) : c(a<sup>2<\/sup> + b<sup>2<\/sup> &#8211; c<sup>2<\/sup>)<\/p>\n<p><strong>Projection rule :<\/strong><br \/>\nIn \u0394ABC<\/p>\n<ul>\n<li>a = b cos C + c cos B,<\/li>\n<li>b = c cos A + a cos C,<\/li>\n<li>c = s cos B + b cos A<\/li>\n<\/ul>\n<p><strong>Mollwiede\u2019s rule :<\/strong><br \/>\nIn \u0394ABC<\/p>\n<ul>\n<li>\\(\\frac{a-b}{c}=\\frac{\\sin \\frac{A-B}{2}}{\\cos \\frac{C}{2}}\\)<\/li>\n<li>\\(\\frac{a+b}{c}=\\frac{\\cos \\frac{A-B}{2}}{\\sin \\frac{C}{2}}\\)<\/li>\n<\/ul>\n<p>Similarly the other two can be written by symmetry.<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 1st Year Maths 1A Properties of Triangles Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p><strong>Tangent rule (or) Napier\u2019s analogy :<\/strong><br \/>\nIn \u0394ABC<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20062\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/Inter-1st-Year-Maths-1A-Properties-of-Triangles-Formulas-1.png\" alt=\"Inter 1st Year Maths 1A Properties of Triangles Formulas 1\" width=\"207\" height=\"186\" \/><\/p>\n<p><strong>Half angle formulae :<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20061\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/Inter-1st-Year-Maths-1A-Properties-of-Triangles-Formulas-2.png\" alt=\"Inter 1st Year Maths 1A Properties of Triangles Formulas 2\" width=\"768\" height=\"352\" \/><\/p>\n<p>cot A = \\(\\frac{\\mathrm{b}^{2}+\\mathrm{c}^{2}-\\mathrm{a}^{2}}{4 \\Delta}\\)<br \/>\ncot B = \\(\\frac{c^{2}+a^{2}-b^{2}}{4 \\Delta}\\)<br \/>\ncot C = \\(\\frac{a^{2}+b^{2}-c^{2}}{4 \\Delta}\\)<\/p>\n<p>\u2192 Area of \u0394ABC 1s given by<\/p>\n<ul>\n<li>\u0394 = \\(\\frac{1}{2}\\)ab sin C = \\(\\frac{1}{2}\\)bc sin A = \\(\\frac{1}{2}\\) ca sin B<\/li>\n<li>\u0394 = \\(\\sqrt{s(s-a)(s-b)(s-c)}\\).<\/li>\n<li>\u0394 = \\(\\frac{a b c}{4 R}\\)<\/li>\n<li>\u0394 = 2R<sup>2<\/sup> sin A sin B sin C<\/li>\n<li>\u0394 = rs<\/li>\n<li>\u0394 = \\(\\sqrt{\\mathrm{rr}_{1} \\mathrm{r}_{2} \\mathrm{r}_{3}}\\)<\/li>\n<\/ul>\n<p>\u2192 If \u2018r\u2019 is radius of in circle and r<sub>1<\/sub>, r<sub>2<\/sub>, r<sub>3<\/sub> are the radii of ex-circles opposite to the vertices A, B, C of \u0394ABC respectively then<br \/>\ni. r = \\(\\frac{\\Delta}{\\mathrm{s}}\\), r<sub>1<\/sub> = \\(\\frac{\\Delta}{\\mathrm{s-a}}\\), r<sub>2<\/sub> = \\(\\frac{\\Delta}{\\mathrm{s-b}}\\), r<sub>3<\/sub> = \\(\\frac{\\Delta}{\\mathrm{s-c}}\\)<\/p>\n<p>\u2192 r = 4R sin\\(\\frac{\\mathrm{A}}{2}\\)sin\\(\\frac{\\mathrm{B}}{2}\\) sin\\(\\frac{\\mathrm{C}}{2}\\)<\/p>\n<ul>\n<li>r<sub>1<\/sub> = 4R sin\\(\\frac{\\mathrm{A}}{2}\\)cos\\(\\frac{\\mathrm{B}}{2}\\)cos\\(\\frac{\\mathrm{C}}{2}\\)<\/li>\n<li>r<sub>2<\/sub> = 4R cos\\(\\frac{\\mathrm{A}}{2}\\)sin\\(\\frac{\\mathrm{B}}{2}\\) cos\\(\\frac{\\mathrm{C}}{2}\\)<\/li>\n<li>r<sub>3<\/sub> = 4R cos\\(\\frac{\\mathrm{A}}{2}\\)cos\\(\\frac{\\mathrm{B}}{2}\\) sin\\(\\frac{\\mathrm{C}}{2}\\)<\/li>\n<\/ul>\n<p>\u2192 r = (s &#8211; a)tan\\(\\frac{A}{2}\\) = (s &#8211; b)tan\\(\\frac{B}{2}\\) = (s &#8211; c)tan\\(\\frac{C}{2}\\)<\/p>\n<ul>\n<li>r<sub>1<\/sub> = s tan\\(\\frac{A}{2}\\) = (s &#8211; b)cot\\(\\frac{C}{2}\\) = (s &#8211; c)cot\\(\\frac{B}{2}\\)<\/li>\n<li>r<sub>2<\/sub> = s tan\\(\\frac{B}{2}\\) = (s &#8211; c)cot\\(\\frac{A}{2}\\) = (s &#8211; a)cot\\(\\frac{c}{2}\\)<\/li>\n<li>r<sub>1<\/sub> = s tan\\(\\frac{A}{2}\\) = (s &#8211; a)cot\\(\\frac{B}{2}\\) = (s &#8211; b)cot\\(\\frac{A}{2}\\)<\/li>\n<\/ul>\n<p>\u2192 \\(\\frac{1}{r}=\\frac{1}{r_{1}}+\\frac{1}{r_{2}}+\\frac{1}{r_{3}}\\)<\/p>\n<p>\u2192 rr<sub>1<\/sub>r<sub>2<\/sub>r<sub>3<\/sub> = \u0394<sup>2<\/sup><\/p>\n<p>\u2192 r<sub>1<\/sub>r<sub>2<\/sub> + r<sub>2<\/sub>r<sub>3<\/sub> + r<sub>3<\/sub>r<sub>1<\/sub> = s<sup>2<\/sup><\/p>\n<p>\u2192 r(r<sub>1<\/sub> + r<sub>2<\/sub> + r<sub>3<\/sub>) = ab + bc + ca &#8211; s<sup>2<\/sup><\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 1st Year Maths 1A Properties of Triangles Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 (r<sub>1<\/sub> &#8211; r)(r<sub>2<\/sub> + r<sub>3<\/sub>) = a<sup>2<\/sup><\/p>\n<p>\u2192 (r<sub>2<\/sub> &#8211; r)(r<sub>3<\/sub> + r<sub>1<\/sub>) = b<sup>2<\/sup><\/p>\n<p>\u2192 (r<sub>3<\/sub> &#8211; r)(r<sub>1<\/sub> + r<sub>2<\/sub>) = c<sup>2<\/sup><\/p>\n<p>\u2192 a = (r<sub>2<\/sub> + r<sub>3<\/sub>)\\(\\sqrt{\\frac{r r_{1}}{r_{2} r_{3}}}\\)<\/p>\n<p>\u2192 b = (r<sub>3<\/sub> + r<sub>1<\/sub>)\\(\\sqrt{\\frac{r r_{2}}{r_{3} r_{1}}}\\)<\/p>\n<p>\u2192 c = (r<sub>1<\/sub> + r<sub>2<\/sub>)\\(\\sqrt{\\frac{r r_{3}}{r_{1} r_{2}}}\\)<\/p>\n<p>\u2192 r<sub>1<\/sub> &#8211; r = 4Rsin<sup>2<\/sup>\\(\\frac{A}{2}\\)<\/p>\n<p>\u2192 r<sub>2<\/sub> &#8211; r = 4Rsin<sup>2<\/sup>\\(\\frac{B}{2}\\)<\/p>\n<p>\u2192 r<sub>3<\/sub> &#8211; r = 4Rsin<sup>2<\/sup>\\(\\frac{C}{2}\\)<\/p>\n<p>\u2192 r<sub>1<\/sub> + r<sub>2<\/sub> = 4R cos<sup>2<\/sup>\\(\\frac{C}{2}\\)<\/p>\n<p>\u2192 r<sub>2<\/sub> + r<sub>3<\/sub> = 4R cos<sup>2<\/sup>\\(\\frac{A}{2}\\)<\/p>\n<p>\u2192 r<sub>3<\/sub> + r<sub>1<\/sub> = 4R cos<sup>2<\/sup>\\(\\frac{B}{2}\\)<\/p>\n<p>\u2192 r<sub>1<\/sub> + r<sub>2<\/sub> + r<sub>3<\/sub> = 4R<\/p>\n<p>\u2192 r + r<sub>2<\/sub> + r<sub>3<\/sub> &#8211; r<sub>1<\/sub> = 4R cos A<\/p>\n<p>\u2192 r + r<sub>1<\/sub> + r<sub>3<\/sub> &#8211; r<sub>2<\/sub> = 4R cos B<\/p>\n<p>\u2192 r + r<sub>1<\/sub> + r<sub>2<\/sub> &#8211; r<sub>3<\/sub> = 4R cos C<\/p>\n<p>\u2192 In an equilateral triangle of side &#8216;a&#8217;<br \/>\narea = \\(\\frac{\\sqrt{3} a^{2}}{4}\\)<br \/>\nR = a\/\u221a3<br \/>\nr = R\/2<br \/>\nr<sub>1<\/sub> = r<sub>2<\/sub> + r<sub>3<\/sub> = 3R\/2<br \/>\nr : R : r<sub>1<\/sub> = 1 : 2 : 3<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 1st Year Maths 1A Properties of Triangles Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p><strong>In circle:<\/strong><br \/>\nThe circle that touches the three sides of a triangle ABC internally is called the \u201cin circle\u201d or inscribed\u201d of its triangle. The centre of the circle is called Incentre denoted by I the radius of the circle is denoted by inradius denoted by Y<\/p>\n<p>\u2192 In a triangle ABC<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20060\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/Inter-1st-Year-Maths-1A-Properties-of-Triangles-Formulas-3.png\" alt=\"Inter 1st Year Maths 1A Properties of Triangles Formulas 3\" width=\"494\" height=\"226\" \/><\/p>\n<p><strong>Excircle:<\/strong><br \/>\nThe circle that touches the side BC (opposite to angle A) internally and the other two sides AB and AC externally is called Excircle. The centre of this circle is called excentre opposite to \u2018A\u2019. denoted by I<sub>1<\/sub>. The radius of this circle is called ex-radius, denoted by r<sub>1<\/sub><\/p>\n<p>|||<sup>ly<\/sup> exradius opposite to angle B is denoted by r<sub>2<\/sub>. The centre of this excircle is denoted by I<sub>2<\/sub> exradius opposite to angle C is denoted by r<sub>3<\/sub>. The centre of this ex-circle is denoted by I<sub>3<\/sub><\/p>\n<p>In a triangle ABC<\/p>\n<ul>\n<li>r<sub>1<\/sub> = \\(\\frac{\\Delta}{s-a}\\)<\/li>\n<li>r<sub>2<\/sub> = \\(\\frac{\\Delta}{s-b}\\)<\/li>\n<li>r<sub>3<\/sub> = \\(\\frac{\\Delta}{s-c}\\)<\/li>\n<\/ul>\n<p>\u2192 r<sub>1<\/sub> = s tan A\/2<\/p>\n<p>\u2192 r<sub>2<\/sub> = s tan B\/2<\/p>\n<p>\u2192 r<sub>2<\/sub> = s tan C\/2<\/p>\n<p>\u2192 r<sub>1<\/sub> = (s &#8211; c)cot\\(\\frac{B}{2}\\)<br \/>\n= (s- b)cot\\(\\frac{C}{2}\\)<\/p>\n<p>\u2192 r<sub>1<\/sub> = (s &#8211; a)cot\\(\\frac{C}{2}\\)<br \/>\n= (s &#8211; c)cot\\(\\frac{A}{2}\\)<\/p>\n<p>\u2192 r<sub>1<\/sub> = (s &#8211; a)cot\\(\\frac{B}{2}\\)<br \/>\n= (s &#8211; c)cot\\(\\frac{A}{2}\\)<\/p>\n<p>\u2192 r<sub>1<\/sub> = \\(\\frac{a}{\\tan \\frac{B}{2}+\\tan \\frac{C}{2}}\\)<\/p>\n<p>\u2192 r<sub>2<\/sub> = \\(\\frac{b}{\\tan \\frac{C}{2}+\\tan \\frac{A}{2}}\\)<\/p>\n<p>\u2192 r<sub>3<\/sub> = \\(\\frac{c}{\\tan \\frac{A}{2}+\\tan \\frac{B}{2}}\\)<\/p>\n<p>\u2192 r<sub>1<\/sub> = 4Rsin\\(\\frac{A}{2}\\)cos\\(\\frac{B}{2}\\)cos\\(\\frac{C}{2}\\)<\/p>\n<p>\u2192 r<sub>2<\/sub> = 4Rcos\\(\\frac{A}{2}\\)sin\\(\\frac{B}{2}\\)cos\\(\\frac{C}{2}\\)<\/p>\n<p>\u2192 r<sub>3<\/sub> = 4Rcos\\(\\frac{A}{2}\\)cos\\(\\frac{B}{2}\\)sin\\(\\frac{C}{2}\\)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Use these Inter 1st Year Maths 1A Formulas PDF Chapter 10 Properties of Triangles to solve questions creatively. Intermediate 1st Year Maths 1A Properties of Triangles Formulas \u2192 Sine Rule : In \u0394ABC = 2R where R is the circumradius of \u0394ABC. \u2192 Cosine Rule : a2 = b2 + c2 &#8211; 2bc. cos A &#8230; <a title=\"Inter 1st Year Maths 1A Properties of Triangles Formulas\" class=\"read-more\" href=\"https:\/\/apboardsolutions.in\/inter-1st-year-maths-1a-properties-of-triangles-formulas\/\" aria-label=\"More on Inter 1st Year Maths 1A Properties of Triangles Formulas\">Read more<\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[14],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/21923"}],"collection":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/comments?post=21923"}],"version-history":[{"count":1,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/21923\/revisions"}],"predecessor-version":[{"id":21924,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/21923\/revisions\/21924"}],"wp:attachment":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/media?parent=21923"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/categories?post=21923"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/tags?post=21923"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}