Tuesday, September 23, 2008

Properties of Triangles and Circles Associated with them - July Dec Revision

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)

R = abc/4Δ

11. Inscribed circle or incircle of a triangle

It is the circle touches each of the sides of the triangle.

The centre of the inscribed circle is the point of intersection of bisectors of the angles of the triangle.

The radius of inscribed circle is denoted by r (it is called in-radius) and it is equal to the length of the perpendicular from its centre to any of the sides of the triangle.

Various formulas that give r.

In- radius ( r )= Δ/s

r = (s-a)tan (A/2) = (s-b) tan (B/2) = (s-c) tan (C/2)

r = [a sin B/2 sin C/2/(Cos A/2)

r = 4R sin (A/2) sin (B/2) sin (C/2)

12. Escribed circles of a triangle

The circle which touches the sides BC and two sides AB and AC produced of a triangle ABC is called the escribed circle opposite to the angle A. Its radius is denoted by r1.

Similarly r2 and r3 denote the radii of the escribed circles opposite to the angles B and C respectively.

The centres of the escribed circles are called the ex-centres.

13. Orthocentre and its distances from the angular points of a triangle

In a Δ ABC, the point at which perpendiculars drawn from the three vertices (heights) meet, it called the ortho centre of the ΔABC

14. Regular polygon and Radii of the inscribed and circumscribing circles of a regular polygon
the centre of the polygon will be the in-centre as well as circumcentre of the polygon.

15. Area of a cyclic quadrilateral

a quadrilateral is a cyclic quadrilateral if its vertices lie on a circle.

Area of cyclic quadrilateral = ½ (ab + cd) sin B

16. Ptolemy’s theorem

In a cyclic quadrilateral ABCD, AC.BD = AB.CD + BC.AD

The product of diagonals is equal to the sum of the products of the lengths of opposite sides.

17. Circum-radius of a cyclic quadrilateral

In a cyclic quadrilateral, the circumcircle of the quadrilateral ABCD is also the circumcircle of Δ ABC.

Past IIT questions

1. The perimeter of a Δ ABC is three times the arithmetic mean of the sines of its angle. If the side a is 1, then the angle a is

a. π/6
b. π/3
c. π/2
d. π

(JEE 1992)

2. If the radius of the circumcircle of an isosceles triangle PQR is equal to PQ = PR, the angle P is

a. π/6
b. π/3
c. π/2
d. 2π/3

(JEE 1992)

3. In a Δ ABC, if (cos A)/a = (cos B)/b = (cos C)/c , and the side a =2, the the area of the triangle is

b. 2
c. (√3)/2
d. √3

(JEE 1993)

4. If in a triangle ABC

2cos A/a = cos B/b + 2 cos C/c = a/bc + b/ca
then the value of the angle A is _____________ degrees

(JEE Screening 1993)

No comments: