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Properties of Triangle Formulae

We will discuss the list of properties of triangle formulae which will help us to solve different types of problems on triangle.

1. The angles of the triangle ABC are denoted by A, B, C and the corresponding opposite sides by a, b, c.

2. s denotes the semi-perimeter of the triangle ABC, βˆ† its area and R the radius of the circle circumscribing the triangle ABC i.e., R is the circum-radius.

3.  asinA = bsinB = csinC = 2R.

4. (i) a = b cos C + c cos B;

(ii) b = c cos A + a cos C, and

(iii) c = a cos B + b cos A.

5. (i) b2 = c2 + a2 - 2ca. cos B or, cos B =  c2+a2βˆ’b22ca

(ii) a2 = b2 + c2 - 2ab. cos A or, cos A = b2+c2βˆ’a22bc

(iii) c2 = a2 + b2 - 2ab. cos C or, cos C = a2+b2βˆ’c22ab


6. (i) tan A = abcR βˆ™ 1b2+c2βˆ’a2

(ii) tan B = abcR βˆ™ 1c2+a2βˆ’b2 and

(iii) tan C = abcR βˆ™ 1a2+b2βˆ’c2.


7. (i) sin A2 = √(sβˆ’b)(sβˆ’c)bc;

(ii) sin B2 = √(sβˆ’c)(sβˆ’a)ca;

(iii) sin C2 = √(sβˆ’a)(sβˆ’b)ab;  


8. (i) cos A2 = √s(sβˆ’a)bc;

(ii) cos BB2 = √s(sβˆ’b)ca;    

(iii) cos C2 = √s(sβˆ’c)ab.


9. (i) tan A2 = √(sβˆ’b)(sβˆ’c)s(sβˆ’a);

(ii) tan B2 = √(sβˆ’c)(sβˆ’a)s(sβˆ’b) and

(iii) tan C2 = √(sβˆ’a)(sβˆ’b)s(sβˆ’c)

10. (i) tan (Bβˆ’C2) = (bβˆ’cb+c) cot A2

(ii) tan (Cβˆ’A2)  = (cβˆ’ac+a) cot B2

(iii) tan (Aβˆ’B2) = (aβˆ’ba+b)  cot C2

10. βˆ† = Β½ Γ— product of lengths of two sides Γ— sine of their included angle 

β‡’ (i) βˆ† = Β½ bc sin A

   (ii) βˆ† = Β½ ca sin B

   (iii) βˆ† = Β½ ab sin C


11. βˆ† = √s(sβˆ’a)(sβˆ’b)(sβˆ’c)

12. R = abc4βˆ†.

13. (i) tan A2 = (sβˆ’b)(sβˆ’c)βˆ†;

(ii) tan B2 = (sβˆ’c)(sβˆ’a)βˆ†and

(iii) tan C2 = (sβˆ’a)(sβˆ’b)βˆ†.


14. (i) cot A2 = s(sβˆ’a)βˆ†;

(ii) cot B2 = s(sβˆ’b)βˆ† and 

(iii) cot C2 = s(sβˆ’c)βˆ†.


15. r = βˆ†s

16. r = 4R sin A2 sin B2 sin C2

17. r = (s - a) tanA2 = (s - b) tanB2 = (s - c) tanC2

i.e., (i) r = (s - a) tanA2

(ii) r = (s - b) tanB2

(iii) r = (s - c) tanC2


18. (i) r1 = βˆ†sβˆ’a

(ii) r1 = βˆ†sβˆ’b

(iii) r1 = βˆ†sβˆ’c


19. r1 = 4R sin A2 cos B2 cos c2

20. r2 = 4R cos A2 sin B2 cos c2

21. r3 = 4R cos A2 cos B2 sin c2

22. (i) r1 = s tanA2

(ii) r1 = s tanB2

(iii) r1 = s tanC2

● Properties of Triangles





11 and 12 Grade Math

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