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Mathematics

Mathematics. Session. Properties of Triangle - 1 . Session Objectives. Session Objective. Sine Rule. Cosine Rule. Projection Rule. Tangents Rule (Napier’s Analogy). Half Angle Formulae. Area of Triangle. Circumcircle of a Triangle and Its Radius . Incircle of a Triangle and Its Radius .

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Mathematics

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  1. Mathematics

  2. Session Properties of Triangle - 1

  3. Session Objectives

  4. Session Objective • Sine Rule • Cosine Rule • Projection Rule • Tangents Rule (Napier’s Analogy) • Half Angle Formulae • Area of Triangle • Circumcircle of a Triangle and Its Radius • Incircle of a Triangle and Its Radius • Excircle of a Triangle and Its Radius

  5. A c A b B C C B a Introduction BC = a CA = b AB = c (b) a + b > c; b + c > a; c + a > b

  6. A B C D Sine Rule In any triangle, sides are proportional to the sines of the opposite angles, i.e. IF A,B,C are in A.P. then B = 60o

  7. A When is an acute angled triangle. B C D b c a Sine Rule Proof :- Draw AD perpendicular to BC

  8. Sine Rule From (i) and (ii), we get c sinB = b sinC Similarly Proved.

  9. Question

  10. Illustrative Problem Solution :

  11. Solution Proved

  12. B C A D c a b Cosine Rule or The Law of Cosines Proof :-

  13. B C A D c a b Cosine Rule or The Law of Cosines

  14. Class Test

  15. In a if Class Exercise - 1 Solution : Let us try to find a,b,c in terms of K.  a = ……….., b = ……...., c = ……….. • b + c = 11k; c + a = 12k; a + b = 13k Adding we get 2 (a + b + c) = 36k

  16. Solution  a + b + c = 18k a = 7k, b = 6k, c = 5k

  17. c = a cosB + b cosA C A B D Projection Rule Proof :- Now c = AB = AD + BD Proved.

  18. Class Test

  19. In any prove that Class Exercise - 2 Solution : LHS =

  20. Solution [Applying projection rule] [Applying cosine rule]

  21. Tangents Rule (Napier’s Analogy) Proof :- By sine rule, we have

  22. Tangents Rule (Napier’s Analogy) Proved

  23. Half Angle Formulae Proof :-

  24. Half Angle Formulae Proved.

  25. Half Angle Formulae

  26. Class Test

  27. Class Exercise - 5 Solution : Given that

  28. Solution  a + b + c = 3b a + c = 2b ... (i) We have to prove

  29. Solution

  30. A B C D Area of a Triangle Proof :- Here BC = a, AB = c, AC = b

  31. Area of a Triangle Proved

  32. (b) Hero’s formula Area of a Triangle Proof :-

  33. Class Test

  34. Class Exercise - 6

  35. Solution  4bc cosA + bc sinA = 4bc  4cosA + sinA = 4

  36. Solution which is not possible, since it in an angle of .

  37. A R E F R O R B C D Circumcircle of a Triangle and its Radius Circum radius = R

  38. Circumcircle of a Triangle and its Radius

  39. A E F r r i r B C D Incircle of a Triangle and its Radius In Radius = r

  40. Incircle of a Triangle and its Radius

  41. A B C I1 Radii of Ex-circle in Terms of Sides and Angles Ex. Radius is = r1

  42. Radii of Ex-circle in Terms of Sides and Angles

  43. Radii of Ex-circle in Terms of Sides and Angles [Proof of these results are same as INCIRCLE and try yourself]

  44. Class Tests

  45. In a if b + c = 3a, then the value of is (a) 1 (b) 2 (c) (d) 3 Class Exercise - 3 Solution :- b + c = 3a = k (sinB + sinC) = 3k sinA

  46. Solution Ans : b

  47. In a if and the side a = 2 units, then area of the is • 1 sq. units (b) 2 sq. units • (c) sq. units (d) sq. units Class Exercise - 4

  48. Solution Given  A = B = C Ans : d

  49. If in a sides a, b, c are in AP, then are in (where are the ex- radii of the ) • AP (b) GP • (c) HP (d) None of these Class Exercise - 7

  50. Solution Since a.b.c are in AP Ans : c  r1, r2, r3 are in H.P

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