Properties & Solution of Triangles

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  • 1. PROPERTIES & SOLUTION OF TRIANGLE
  • 2. SINE RULE
  • 3. EXAMPLES
  • 4. COSINE FORMULA b2 c2 a 2 cos A = or 2bc a² = b² + c² - 2bc cos A = b2 + c2 + 2bc cos (B + C) c2 a 2 b2 cos B = 2 ca a2 b 2 c2 cos C = 2a b
  • 5. EXAMPLES
  • 6. PROJECTION FORMULA a = b cosC + c cosB b = c cosA + a cosC c = a cosB + b cosA
  • 7. EXAMPLES
  • 8. NAPIER’S ANALOGY - TANGENT RULE B C b c A tan cot 2 b c 2 C A c a B tan cot 2 c a 2 A B a b C tan cot 2 a b 2
  • 9. EXAMPLES
  • 10. TRIGONOMETRIC FUNCTIONS OF HALF ANGLES
  • 11. AREA OF TRIANGLE (∆)
  • 12. EXAMPLES
  • 13. M-N RULE m cot n cot n cot B m cot C
  • 14. EXAMPLES
  • 15. RADIUS OF CIRCUMCIRCLE
  • 16. EXAMPLES
  • 17. RADIUS OF THE INCIRCLE
  • 18. RADIUS OF THE EX- CIRCLES
  • 19. EXAMPLES
  • 20. LENGTH OF ANGLE BISECTORS, MEDIANS &ALTITUDES 2 bc cos A 2 b c 1 2 b2 2 c2 a2 2
  • 21. EXAMPLES
  • 22. THE DISTANCES OF THE SPECIAL POINTS FROMVERTICES AND SIDES OF TRIANGLE
  • 23. EXAMPLES
  • 24. ORTHOCENTER AND PEDAL TRIANGLE The triangle KLM which is formed by joining the feet of the altitudes is called the Pedal Triangle. Its angles are - 2A, - 2B and - 2C. Its sides are a cosA = R sin 2A, b cosB = R sin 2B and c cosC = R sin 2C Circumradii of the triangles PBC, PCA, PAB and ABC are equal.
  • 25. EXCENTRAL TRIANGLEThe triangle formed by joining the three excentres I1, I2 andI3 of D ABC is called the excentral or excentric triangle.
  • 26.
  • 27. DISTANCE BETWEEN SPECIAL POINTS
  • 28. EXAMPLES