TS Inter 1st Year Maths 1A Properties of Triangles Formulas

Learning these TS Inter 1st Year Maths 1A Formulas Chapter 10 Properties of Triangles will help students to solve mathematical problems quickly.

TS Inter 1st Year Maths 1A Properties of Triangles Formulas

→ In any ΔABC, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\) = 2R where a, b, c are the lengths of sides BC, CA and AB of a triangle ABC. A, B, C are the angles at the vertices of ΔABC, and R is the circum radius of the ΔABC. This is called the SINE RULE.

→ In any ΔABC,

  • a2 = b2 + c2 – 2bc cos A
  • b2 = c2 + a2 – 2ca cos B
  • c2 = a2 + b2 – 2ab cos C is the COSINE RULE.

→ The angles A, B, C can be found by the formulae

  • cos A = \(\frac{b^2+c^2-a^2}{2 b c}\)
  • cos B = \(\frac{c^2+a^2-b^2}{2 c a}\)
  • cos C = \(\frac{a^2+b^2-c^2}{2 a b}\)

→ In any ΔABC,

  • a = b cos C + c cos C
  • b = a cos C + c cos A
  • c = a cos B + b cos A (projection formulae)

→ tan\(\left(\frac{B-C}{2}\right)=\frac{b-c}{b+c}\) cot\(\frac{A}{2}\) (or) tan\(\left(\frac{c-A}{2}\right)=\frac{c-a}{c+a}\)cot\(\frac{B}{2}\) (or)
tan\(\left(\frac{A-B}{2}\right)=\frac{a-b}{a+b}\) cot \(\frac{C}{2}\) is the Napier’s analogy

→ If a + b + c = 2s which is the perimeter of ΔABC then
sin\(\frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{b c}}\), sin\(\frac{B}{2}=\sqrt{\frac{(s-a)(s-c)}{a c}}\) and sin\(\frac{c}{2}=\sqrt{\frac{(s-a)(s-b)}{a b}}\)
cos\(\frac{A}{2}=\sqrt{\frac{s(s-a)}{b c}}\), cos\(\frac{\mathrm{B}}{2}=\sqrt{\frac{s(s-b)}{a c}}\), cos \(\frac{c}{2}=\sqrt{\frac{s(s-c)}{a b}}\) and
tan\(\frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\), tan\(\frac{B}{2}=\sqrt{\frac{(s-a)(s-c)}{s(s-b)}}\), tan\(\frac{c}{2}=\sqrt{\frac{(s-a)(s-b)}{s(s-c)}}\)

TS Inter 1st Year Maths 1A Properties of Triangles Formulas

→ Area of ΔABC Δ = \(\frac{1}{2}\)bc sin A = \(\frac{1}{2}\)ca sin B = \(\frac{1}{2}\)ab sin C
= \(\sqrt{s(s-a)(s-b)(s-c)}=\frac{a b c}{4 R}\) = 2R sin A sin B sin C

→ If ‘r’ is the radius of incircle of ΔABC; r1, r2, r3 are the radii of excircle then
r = \(\frac{\Delta}{s}\), r1 = \(\frac{\Delta}{s-a}\), r2 = \(\frac{\Delta}{s-b}\) and r3 = \(\frac{\Delta}{s-c}\)

→ Also r = 4R sin\(\frac{A}{2}\)sin\(\frac{B}{2}\)sin\(\frac{C}{2}\)

  • r1 = 4R sin\(\frac{A}{2}\)cos\(\frac{B}{2}\)cos\(\frac{C}{2}\)
  • r2 = 4R sin\(\frac{B}{2}\) cos\(\frac{C}{2}\) cos\(\frac{A}{2}\)
  • r3 = 4R sin\(\frac{C}{2}\) cos\(\frac{A}{2}\) cos\(\frac{B}{2}\)

→ In any ΔABC, \(\frac{a+b}{c}=\frac{\cos \left(\frac{A-B}{2}\right)}{\sin \frac{C}{2}}\)
\(\frac{b+c}{a}=\frac{\cos \left(\frac{B-C}{2}\right)}{\sin \frac{A}{2}}\)
and \(\frac{c+a}{b}=\frac{\cos \left(\frac{C-A}{2}\right)}{\sin \frac{B}{2}}\)

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