Triangles

Triangle
A closed figure bounded by three line segments is called a triangle. It is represented by the symbol ∆. It has six parts, three angles and three sides.

A ∆ABC has
  1. three vertices (A,B and C)
  2. three sides (AB, BC and CA)
  3. Three angles (∠A, ∠B and ∠C)

Classification of triangles according to sides

Equilateral Triangle

Equilateral triangle

A triangle having all sides equal is called an equilateral triangle.

The measure of each angles of an equilateral triangle is 60°. When all the angles are equal then all the sides are also equal. So, in the left triangle ABC,
AB = BC = CA
∠A = ∠B = ∠C = 60°
Isosceles Triangle

Isosceles Triangle

A triangle having two sides equal is called an isosceles triangle.

In isosceles triangle the angles opposite to equal sides are equal.So,
AC = BC
∠B = ∠A

Isosceles Triangle

A triangle having all sides of different lengths is called a scalene triangle.

In scalene triangle
AB ≠ BC ≠ CA
∠C ≠ ∠B ≠ ∠A

Classification of triangles according to angles

Right angled triangle

A triangle having one angle equal to 90° is called a right angled triangle or a right triangle.

In a right angled triangle, the side opposite to the right triangle is called its hypotenuse and the remaining two sides are called its legs. One of the legs is called base and other perpendicular.

In the left triangle ABC,
∠B = 90°, AC is hypotenuse and AB and BC are legs. BC is base and AB is perpendicular.
Obtuse angled triangle

Obtuse Angled Triangle

A triangle having one angle greater than 90° and less than 180° is called a obtuse angled triangle.

Acute Angled Triangle

A triangle having all the angles less than 90° is called an acute angled triangle.

Terms related to a triangle


centroid

Medians of a triangle

A line segment joining the vertex to the mid-point of the opposite side of a triangle is called a median. A triangle has three medians. All the three medians intersect at a point.
D is the mid-point of the side BC of ∆ABC. AD is the median of the triangle ∆ABC.

Centroid

The point of intersection of all the medians of a triangle is called its centroid.
D, E and F are the mid-point of the side BC, CA and AB of ∆ABC. G is the centroid of the triangle ∆ABC.

orthocenter

Altitudes of a triangle

A line segment joining the vertex to a point on the opposite side of a triangle such that the line segment is perpendicular to the opposite side is called the altitude of the triangle. The opposite side is called the base. A triangle has three altitudes. All the three altitudes intersect at a point.
AD is the altitude of the triangle ∆ABC.

Orthocenter

The point of intersection of all the altitudes of a triangle is called its orthocenter.
AD, BE and CF are the altitudes of ∆ABC. O is the orthocenter of the triangle ∆ABC.

Incenter

Incenter of a tiangle

The point of intersection of all the internal bisectors of the angles of a triangle is called its incenter.
I is the incenter of the triangle ∆ABC.


circum center

Circum-center of a tiangle

The point of intersection of all the perpendicular bisectors of the sides of a triangle is called its circum-center.
O is the circum-center of the triangle ∆ABC. OA = OB = OC = radius of the circumcircle.

    Properties of triangles
  • The sum of angles of a triangle is 180°.
  • In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the remaining two sides.
  • If two sides of a triangle are equal in length, then the angles opposite to them are equal.
  • If two angles of a triangle are equal then the sides opposite to them are of equal length.
  • The sum of any two sides of a triangle is always greater than the third side.
  • The difference of any two sides of a triangle is less than the third side.
  • If two sides of a triangle are of unequal lengths then the greater side has the greater angle opposite to it.
  • If two angles of a triangle are unequal then the greater angle has the greater side opposite to it.


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