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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, β 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+a2βb22ca
(ii) a2 = b2 + c2 - 2ab. cos A or, cos A = b2+c2βa22bc
(iii) c2 = a2 + b2 - 2ab. cos C or, cos C = a2+b2βc22ab
6. (i) tan A = abcR β 1b2+c2βa2
(ii) tan B = abcR β 1c2+a2βb2 and
(iii) tan C = abcR β 1a2+b2βc2.
7. (i) sin A2 = β(sβb)(sβc)bc;
(ii) sin B2 = β(sβc)(sβa)ca;
(iii) sin C2 = β(sβa)(sβb)ab;
8. (i) cos A2 = βs(sβa)bc;
(ii) cos BB2 = βs(sβb)ca;
(iii) cos C2 = βs(sβc)ab.
9. (i) tan A2 = β(sβb)(sβc)s(sβa);
(ii) tan B2 = β(sβc)(sβa)s(sβb) and
(iii) tan C2 = β(sβa)(sβb)s(sβc)
10. (i) tan (BβC2) = (bβcb+c) cot A2
(ii) tan (CβA2) = (cβac+a) cot B2
(iii) tan (AβB2) = (aβba+b) cot C2
10. β = Β½ Γ product of lengths of two sides Γ sine of their
included angle
β (i) β = Β½ bc sin A
(ii) β = Β½ ca sin B
(iii) β = Β½ ab sin C
11. β = βs(sβa)(sβb)(sβc)
12. R = abc4β.
13. (i) tan A2 = (sβb)(sβc)β;
(ii) tan B2 = (sβc)(sβa)βand
(iii) tan C2 = (sβa)(sβb)β.
14. (i) cot A2 = s(sβa)β;
(ii) cot B2 = s(sβb)β and
(iii) cot C2 = s(sβc)β.
15. r = βs
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 = βsβa
(ii) r1 = βsβb
(iii) r1 = βsβc
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
11 and 12 Grade Math
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