Propeties of-triangles
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Propeties of-triangles

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Propeties of-triangles

Propeties of-triangles

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  • 1. AIEEEportal.com Properties of Triangles ( by tarun gehlot) 1. The perpendicular bisectors of the sides of a triangle are concurrent. The point of concurrence is called circumcentre of the triangle. If S is the circumcentre of ΔABC, then SA = SB = SC. The circle with center S and radius SA passes through the three vertices A, B, C of the triangle. This circle is called circumcircle of the triangle. The radius of the circumcircle of ΔABC is called circumradius and it is denoted by R. 2. Sine Rule : a b c = = = 2R. sin A sin B sin C ∴ a = 2R sin A, b = 2R sin B, c = 2R sin C. 3. Cosine Rule : a2 = b2 + c2 – 2bc cos A, b2 = c2 + a2 – 2ca cos B, c2 = a2 + b2 – 2ab cos C. 4. cos A = b2 + c 2 − a 2 2bc , cos B = cos C = a2 + b2 − c 2 2ab . c 2 + a2 − b2 2ca , 5. Projection Rule : a = b cos C + c cos B, b = c cos A + a cos C, c = a cos B + b cos A. 6. Tangent Rule or Napier’s Analogy : tan⎛ ⎜ A B−C⎞ b−c cot , ⎟= 2 2 ⎠ b+c ⎝ B ⎛C−A ⎞ c −a cot , tan⎜ ⎟= 2 ⎝ 2 ⎠ c+a C ⎛ A −B⎞ a −b tan⎜ cot . ⎟= 2 ⎝ 2 ⎠ a+b 7. Mollweide Rule : a+b = c ⎛ A −B⎞ ⎛ A −B⎞ cos⎜ sin⎜ ⎟ ⎟ 2 ⎠ a−b ⎝ ⎝ 2 ⎠ = , C C c sin cos 2 2 (s − b)(s − c ) B , sin = bc 2 8. sin A = 2 9. cos A = 2 s( s − a) B , cos = bc 2 10. tan A = 2 ( s − b)(s − c ) s(s − a) , tan (s − c )(s − a) C , sin = ca 2 s(s − b) C , cos = ca 2 B = 2 (s − c )(s − a) s( s − b) (s − a)(s − b) . ab s( s − c ) . ab , tan 1 C = 2 (s − a)(s − b) s(s − c )
  • 2. Properties of Triangles 11. tan A Δ ( s − b)(s − c ) = = , 2 s(s − a) Δ tan B Δ (s − c )(s − a) , = = Δ 2 s( s − b) tan C Δ (s − a)(s − b) = = Δ 2 s(s − c ) 12. cot . A s(s − a) B s(s − b) C s(s − c ) , cot = , cot = = 2 2 Δ Δ Δ 2 13. Area of ΔABC is Δ = 1 1 1 2 bc sin A = ca sin B = sin C = 2R sin A sin B sin C = 2 2 2 abc s( s − a)(s − b)(s − c ) . 4R 14. r = B C Δ A A B C = (s − a) tan = (s − b) tan = (s − c ) tan = = 4R sin sin sin 2 2 2 2 2 2 s a cot B + cot C 2 C = cot A + cot 2 B 2 15. r1 = Δ A B C A B C = 4R sin cos cos = s tan = (s − b) cot = (s − c ) cot = s−a 2 2 2 2 2 2 16. r2 = Δ C A B = = s tan = (s − c ) cot = (s − a) cot 2 2 2 s−b 4R cos 17. r3 = A B C sin cos = 2 2 2 b tan A C + tan 2 2 . A B C Δ = = s tan = (s − a) cot = (s − b) cot 2 2 2 s−c c . B A tan + tan 2 2 18. 1 1 1 1 + + = . r1 r2 r3 r 19. r r1 r2 r3 = Δ2. ∑ a sin(B − C) = 0 . ii) ∑ a cos(B − C) = 3abc 20. i) 3 3 iii) a2 sin 2B + b2 sin 2A = 4Δ 2 a . C B tan + tan 2 2 2 b = cot C 2 + cot A 2
  • 3. Properties of Triangles a2 + b2 + c 2 4Δ 21. i) cotA + cotB + cotC = ii) cot A B C (a + b + c )2 . cot cot = 2 2 2 4Δ 22. i) If a cos B = b cos A, then the triangle is isosceles. ii) If a cos A = b cos B, then the triangle is isosceles or right angled. iii)If a2 + b2 + c2 = 8R2, then the triangle is right angled. iv) If cos2A + cos2B + cos2C = 1, then the triangle is right angled. v) If cosA = vi) If sin B , then the triangle is isosceles. 2 sin C a b c , then the triangle is equilateral. = = cos A cos B cos C vii) If cosA + cosB + cosC = 3/2, then the triangle is equilateral. viii) If sinA + sinB + sinC = 3 3 , then the triangle is equilateral. 2 ix) If cotA + cotB + cotC = 3 , then the triangle is equilateral. 23. i) If a2 + b2 a −b 2 2 = sin( A + B) , then C = 90°. sin( A − B) ii) If a+b b = 1, then C = 60°. + b+c c+a iii)If 1 1 3 , then A = 60° + = a+b a+c a+b+c iv) If b a2 − c 2 + c a2 − b 2 = 0, then A = 60°. C B A are in H.P. , sin2 , sin2 2 2 2 i) a, b, c are In H.P. ⇔ sin2 ii) a, b, c are in A.P. ⇔ cot B A C , cot , cot 2 2 2 iii) a, b, c are in A.P. ⇔ tan A B C are in H.P. , tan , tan 2 2 2 iv) a2, b2, c2 are in A.P. ⇔ cotA, cotB, cotC are in A.P. v) a2, b2, c2 are in A.P. ⇔ are in A.P. tanA, tanB, tanC are in H.P AIEEEportal.com 3