Revision Ch.34 Properties of Triangles and circles - 1

1. Semiperimeter of a triangle is denoted by s.
2. Area of a triangle is denoted by Δ or S.
3. a,b, and c represent sides BC,CA, and AB

4. Sine rule

In any Δ ABC
Sin A/a = Sin B/b = Sin C/c

5. Cosine Formulae

In any Δ ABC

Cos A = [b² + c² -a²]/2bc


Cos B = [c² +a² –b²]/2ac


Cos C = [a² + b² –c²]/2ab

6. Projection formulae

In any Δ ABC

a = b Cos C + C cos B

b= c Cos A + A Cos C

c = a Cos B + b cos A


7. trigonometrical ratios of half of the angles of a triangle

1. Sin A/2 = √[(s-b)(s-c)/bc]

2. Cos A/2 = √[s(s-a)/bc]

3. tan A/2 = √(s-b)(s-c)/s(s-a)]



8. Area of a triangle

S = ½ ab Sin C = ½ bc sina = ½ ac sin B


9. Napier’s analogy

In any triangle ABC

Tan [(b-c)/2] = [(b-c)cot (A/2)]/(b+c)

10. Circumcircle of a triangle

The circle which passes through the angular points or vertices of a triangle ABC is called its circumcircle.

The centre of this circle can be found by locating the point of intersection of perpendicular bisectors of the sides. It is called circumcentre.

The circumcentre may lie within, outside or upon one of the sides of the triangle.

In a right angled triangle the cicumcentre is vertex where right angle is formed.

The radius of circumcircle is denoted by R.

R = a/(2 Sin A) = b/(2 sin B) = c/(2 sin C)