Properties & Solution of Triangles

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Properties & Solution of Triangles

  1. 1. PROPERTIES & SOLUTION OF TRIANGLE
  2. 2. SINE RULE
  3. 3. EXAMPLES
  4. 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. 5. EXAMPLES
  6. 6. PROJECTION FORMULA a = b cosC + c cosB b = c cosA + a cosC c = a cosB + b cosA
  7. 7. EXAMPLES
  8. 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. 9. EXAMPLES
  10. 10. TRIGONOMETRIC FUNCTIONS OF HALF ANGLES
  11. 11. AREA OF TRIANGLE (∆)
  12. 12. EXAMPLES
  13. 13. M-N RULE m cot n cot n cot B m cot C
  14. 14. EXAMPLES
  15. 15. RADIUS OF CIRCUMCIRCLE
  16. 16. EXAMPLES
  17. 17. RADIUS OF THE INCIRCLE
  18. 18. RADIUS OF THE EX- CIRCLES
  19. 19. EXAMPLES
  20. 20. LENGTH OF ANGLE BISECTORS, MEDIANS &ALTITUDES 2 bc cos A 2 b c 1 2 b2 2 c2 a2 2
  21. 21. EXAMPLES
  22. 22. THE DISTANCES OF THE SPECIAL POINTS FROMVERTICES AND SIDES OF TRIANGLE
  23. 23. EXAMPLES
  24. 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. 25. EXCENTRAL TRIANGLEThe triangle formed by joining the three excentres I1, I2 andI3 of D ABC is called the excentral or excentric triangle.
  26. 26.
  27. 27. DISTANCE BETWEEN SPECIAL POINTS
  28. 28. EXAMPLES

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