Solutions / Properties of triangles for IIT-JEE MAINS /ADVANCED

Video lectures  of KAMAL SIR  for  Properties of triangles

 

 

QUICK   REVIEW

In a triangle ABC the angles are denoted by capital letters A, B and C and the length of the sides opposite to these angles are denoted by small letters a, b and c. Semi perimeter of the triangle is
given by s where  2s = a + b+c  and its area is denoted by Δ

SINE  RULE

Sine rule is an  important tool that relates   side lengths with angles of triangles  and circum radius of triangle

PROPERTIES OF TRIANGLE 1

NAPIER’S ANALOGY (TANGENT RULE)

PROPERTIES OF TRIANGLE

COSINE RULE
In a triangle ABC

properties of triangle -- cosine rule
properties of triangle — cosine rule

 

PROJECTION RULE

(i) a = b cos C + c cos B (ii) b = c cos A + a cos C (iii) c = a cos B + b cos C

HALF ANGLE FORMULAE

PROPERTIES OF TRIANGLE

 

.m-n THEOREM

PROPERTIES OF TRIANGLES

PROPERTIES OF TRIANGLES

 

CENTROID AND MEDIANS OF A TRIANGLE
The line joining any vertex of a triangle to the mid point of the opposite side of the triangle is called the median of the triangle. The three medians of a triangle are concurrent and the point of concurrency of the medians of any triangle is called the centroid of the triangle. The centroid divides the median
in the ratio 2 : 1.

 

Circum circle

PROPERTIES OF TRIANLES

The circle which passes through the angular points of a ABC, is called its circumcircle. The centre of this circle i.e., the point of concurrency of the perpendicular bisectors of the sides of the ABC, is called the circumcenter.

Radius of the circumcircle is given by the following formulae

PROPERTIES OF TRIANGLE
ORTHOCENTER AND PEDAL TRIANGLE OF A TRIANGLE.
In a triangle the altitudes drawn from the three vertices to the opposite sides are concurrent and the point of cuncurrency of the altitudes of the triangle is called the orthocenter of the triangle. The triangle formed by joining the feet of these perpendiculars is called the pedal triangle i.e.
DEF is the pedal triangle of ABC.

PROPERTIES OF TRIANGLES

 

BISECTORS OF THE ANGLES
If AD bisects the angle A and divide the base into portions x and y, we have, by Geometry,
properties of triangle

 

INCIRCLE
the circle which can be inscribed within the triangle so as to touch each of the sides of the triangle is called its incircle. The centre of this circle i.e., the point of concurrency of angle bisectors of the triangle is called the incentre of the ABC

properties of triangle

The distances Between the special points
properties of triangle

ESCRIBED CIRCLES
The circle which touches the side BC and the two sides AB and AC produced is called the escribed circle opposite the angle A. Its centre and radius will be denoted by I1 and r1 respectively.

properties of triangle

Excentral triangle
The triangle formed by joining the three excentres I1, I2 and I3 of
DABC is called the excentral or excentric triangle. Not that

properties of triangles

Inscribed & Circumscribed Polygons
(Important Formulae)

sot 46

 

SOLUTION OF TRIANGLES
When any three of the six elements (except all the three angles) of a triangle are given, the triangle is known completely. This process is called the solution of triangles.

sot 47

sot 48

 

Leave a Reply

Your email address will not be published. Required fields are marked *