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, ∆ its area and R the radius of the circle circumscribing the triangle ABC i.e., R is the circum-radius.

3.  \(\frac{a}{sin A}\) = \(\frac{b}{sin B}\) = \(\frac{c}{sin C}\) = 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) b\(^{2}\) = c\(^{2}\) + a\(^{2}\) - 2ca. cos B or, cos B =  \(\frac{c^{2} + a^{2} - b^{2}}{2ca}\)

(ii) a\(^{2}\) = b\(^{2}\) + c\(^{2}\) - 2ab. cos A or, cos A = \(\frac{b^{2} + c^{2} - a^{2}}{2bc}\)

(iii) c\(^{2}\) = a\(^{2}\) + b\(^{2}\) - 2ab. cos C or, cos C = \(\frac{a^{2} + b^{2} - c^{2}}{2ab}\)

6. (i) tan A = \(\frac{abc}{R}\) ∙ \(\frac{1}{b^{2} + c^{2} - a^{2}}\)

(ii) tan B = \(\frac{abc}{R}\) ∙ \(\frac{1}{c^{2} + a^{2} - b^{2}}\) and

(iii) tan C = \(\frac{abc}{R}\) ∙ \(\frac{1}{a^{2} + b^{2} - c^{2}}\).

7. (i) sin \(\frac{A}{2}\) = \(\sqrt{\frac{(s - b)(s - c)}{bc}}\);

(ii) sin \(\frac{B}{2}\) = \(\sqrt{\frac{(s - c)(s - a)}{ca}}\);

(iii) sin \(\frac{C}{2}\) = \(\sqrt{\frac{(s - a)(s - b)}{ab}}\);  

8. (i) cos \(\frac{A}{2}\) = \(\sqrt{\frac{s(s - a)}{bc}}\);

(ii) cos B\(\frac{B}{2}\) = \(\sqrt{\frac{s(s - b)}{ca}}\);    

(iii) cos \(\frac{C}{2}\) = \(\sqrt{\frac{s(s - c)}{ab}}\).

9. (i) tan \(\frac{A}{2}\) = \(\sqrt{\frac{(s - b)(s - c)}{s(s - a)}}\);

(ii) tan \(\frac{B}{2}\) = \(\sqrt{\frac{(s - c)(s - a)}{s(s - b)}}\) and

(iii) tan \(\frac{C}{2}\) = \(\sqrt{\frac{(s - a)(s - b)}{s(s - c)}}\)

10. (i) tan (\(\frac{B - C}{2}\)) = (\(\frac{b - c}{b + c}\)) cot \(\frac{A}{2}\)

(ii) tan (\(\frac{C - A}{2}\))  = (\(\frac{c - a}{c + a}\)) cot \(\frac{B}{2}\)

(iii) tan (\(\frac{A - B}{2}\)) = (\(\frac{a - b}{a + b}\))  cot \(\frac{C}{2}\)

10. ∆ = ½ × product of lengths of two sides × sine of their included angle 

⇒ (i) ∆ = ½ bc sin A

   (ii) ∆ = ½ ca sin B

   (iii) ∆ = ½ ab sin C

11. ∆ = \(\sqrt{s(s - a)(s - b)(s - c)}\)

12. R = \(\frac{abc}{4∆}\).

13. (i) tan \(\frac{A}{2}\) = \(\frac{(s - b)(s - c)}{∆}\);

(ii) tan \(\frac{B}{2}\) = \(\frac{(s - c)(s - a)}{∆}\)and

(iii) tan \(\frac{C}{2}\) = \(\frac{(s - a)(s - b)}{∆}\).

14. (i) cot \(\frac{A}{2}\) = \(\frac{s(s - a)}{∆}\);

(ii) cot \(\frac{B}{2}\) = \(\frac{s(s - b)}{∆}\) and 

(iii) cot \(\frac{C}{2}\) = \(\frac{s(s - c)}{∆}\).

15. r = \(\frac{∆}{s}\)

16. r = 4R sin \(\frac{A}{2}\) sin \(\frac{B}{2}\) sin \(\frac{C}{2}\)

17. r = (s - a) tan\(\frac{A}{2}\) = (s - b) tan\(\frac{B}{2}\) = (s - c) tan\(\frac{C}{2}\)

i.e., (i) r = (s - a) tan\(\frac{A}{2}\)

(ii) r = (s - b) tan\(\frac{B}{2}\)

(iii) r = (s - c) tan\(\frac{C}{2}\)

18. (i) r\(_{1}\) = \(\frac{∆}{s - a}\)

(ii) r\(_{1}\) = \(\frac{∆}{s - b}\)

(iii) r\(_{1}\) = \(\frac{∆}{s - c}\)

19. r\(_{1}\) = 4R sin \(\frac{A}{2}\) cos \(\frac{B}{2}\) cos \(\frac{c}{2}\)

20. r\(_{2}\) = 4R cos \(\frac{A}{2}\) sin \(\frac{B}{2}\) cos \(\frac{c}{2}\)

21. r\(_{3}\) = 4R cos \(\frac{A}{2}\) cos \(\frac{B}{2}\) sin \(\frac{c}{2}\)

22. (i) r\(_{1}\) = s tan\(\frac{A}{2}\)

(ii) r\(_{1}\) = s tan\(\frac{B}{2}\)

(iii) r\(_{1}\) = s tan\(\frac{C}{2}\)

● Properties of Triangles

• The Law of Sines or The Sine Rule
• Theorem on Properties of Triangle
• Projection Formulae
• Proof of Projection Formulae
• The Law of Cosines or The Cosine Rule
  
• Area of a Triangle
• Law of Tangents
• Properties of Triangle Formulae
• Problems on Properties of Triangle

11 and 12 Grade Math

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