TRIANGLE

TRIANGLE

Definition : A closed figure formed by three sides.

Ø A, B and C are vertices and a, b, c are sides of triangle ABC.

Ø Sum of two sides of a triangle is always greater than third side of that triangle.

(i) a+b>c                              (ii) b+c>a                        (iii) c+a>b

Ø Difference of two sides of a triangle is always smaller than the third side of that triangle.

Ø Theorem : sum of all interior angle of a triangle is 180°.

Proof :

Construction : Drawing a line parallel to BC through point A.


Theorem : The exterior angle of a triangle is equal sum of two opposite interior angles.


Types of Triangle :-

(i) On the basis of sides-

(a) Equilateral Triangle :- A triangle which has all three sides equal is called equilateral triangle.

(b) Isosceles Triangle :- A triangle which has two equal sides is called isosceles triangles.

(c) Scalene Triangle :- A triangle which has three different sides is called scalene triangle.

(ii) On the basis of Angles :-

(a) Acute Angle Triangle :- A triangle which have all angles less than 90° is called acute angle triangle.

Ø  Also, in acute angle triangle 

Where c is longest side of triangle.

(b) Right Angle Triangle :- A triangle in which one of angles is 90° is called right angle triangle.

Ø Also in Right angle triangle.

Where C is the longest side.

(a)  Obtuse Angle Triangle- A triangle in which one of angles is obtuse angle  is called obtuse angle triangle.

Ø Also in obtuse angle triangle.

Where C is the longest side.

·        Relationship of sides to interior angles in a triangles.

Ø The shortest side is always opposite the smallest interior angle.

Ø The long side is always opposite the largest interior angle.

Ø  

        

CENTRES OF TRIANGE

1. Centroid (G) : Intersection point of three medians is called centroid.

·        Median :- Line  joining from vertex to midpoint of opposite side is called median.

In above triangle, AD, BE and CF are medians and G is the centriod of .

·        Centroid divides the median in the ratio 2 : 1.

AG : GD =BG : GE = CG : GF =2 : 1

·        Median divides area of triangle in two equal points.

·        Centroid always forms inside in all types of triangle.

2 Incenter (I): Intersection point of angle bisectors of all interior angles in a triangle is called incentre.

Incenter is the center of largest circle. That will fit inside the triangle and touch all three sides.

It is always inside the triangle.

Angles on Incenter:-

 

Proof :

 

Inradius (r):

3 Circumcenters:- Intersection point of perpendicular bisectors of three sides of a triangle, is called circumcentre.

Ø ‘C’ is the circumcenter of It is equidistant from all three vertices.

Ø The distance between circumcenter and one of vertices of triangle is called circumradius.

Ø If a circle is drawn, taking ‘C’ as a centre and ‘R’ as radius. Then it will pass through three vertices of triangle. This circle is called circum circle of triangle.

Ø Position of circum center in different triangles.

(1)  In Acute angle triangle.

Ø Circumcenter lies inside the triangle.

(2)  In obtuse-angle triangle: It lies outside the triangle.

(3)  In Right angle triangle:- It lies on the midpoint of hypotenuse.

4 Orthocenter:-

                   The point where the three attitudes of a triangle intersect is called orthocenter.

Altitude:- An Altitude is a line which passes through a vertex of triangle and meets the opposite side at right angle.

                   A triangle has three altitudes.

Ø AD, BE and CF are altitudes and ‘O’ is orthocenter of 

Ø Position of Orthocenter in different triangle:-

1. In Acute Angle triangle : Orthocenter lies inside the triangle.

2. In obtuse Angle Triangle:- Orthocenter lies outside the triangle.

3 In Right angle Triangle :- Orthocenter lies on the vertex, where 90° angle is formed.

Proof :-