Class 8: Triangles (Lecture Notes I)

Triangle: We just studied polygons. Triangle is a polygon with three sides. So, we could define a triangle as a plane closed figure bounded by three line segments.

A triangle is a polygon with three edges and three vertices. It is one of the most basic shapes in geometry. A triangle with vertices A, B, and C is denoted by ∆ABC.

Kind of Triangles

Classification of triangles based on the length of the sides

Scalene Triangle

A triangle in which all three sides are of different lengths is called Scalene Triangle.

In this type of triangle:

$\angle A \neq \angle B \neq \angle C$

Isosceles Triangle

A triangle in which two sides are of the same length is called Isosceles Triangle

In this type of triangle:

$\angle B = \angle C$

Equilateral Triangle

A triangle in which all three sides are of the same length is called Equilateral Triangle.

In this type of triangle:

$\angle A = \angle B = \angle C$

Some of the diagrams have been adopted from https://en.wikipedia.org/wiki/Triangle

Classification of Triangles based on the angles

Acute-angled Triangle	A triangle in which all the three angles are more than $0^{\circ}$ and less than $90^{\circ}$ is called acute-angled triangle.
Right-angles Triangle	A triangle in which one of the angles is $90^{\circ}$ is called right-angled triangle.
Obtuse-angled triangle	A triangle in which one of the angles is more than $90^{\circ}$ but less than $180^{\circ}$ is called obtuse-angled triangle.

Some of the diagrams have been adopted from https://en.wikipedia.org/wiki/Triangle

Term related to a Triangle

Median	A line segment joining the vertex to the mid-point of the opposite side of a triangle is called median. In this vertex $A$ is meeting at point $D$ (such that $BD=DC$ ) midpoint of $BC$
Centroid	The point of intersection of three medians is called centroid.
Altitude	The perpendicular drawn from the vertex to the opposite side. Here $AD$ is the altitude of the triangle $AD \ and\ BC$ is the base.
Orthocenter	The intersection of the three altitudes is called the Orthocenter of the triangle. Here $A$ is the Orthocenter of the triangle.
Angle Bisector	A line segment that bisects and interior angle of a triangle is called angle bisector. Here $AD$ is bisecting $\angle BAC$ into two equal $\angle BAD \ and\ \angle DAC$
Incentre and Incircle	The point of intersection of internal angle bisectors is called the Incentre. $I$ is the Incentre of the triangle. Now if you draw a circle with the center $I$ in such a way that it touches all the three sides of the triangle, then that is called Incircle.
Perpendicular Bisector or Right Bisector	A line bisecting any side of the triangle and perpendicular to it is called perpendicular bisector of that side of the triangle. Here BC is being bisected by $DE. BD=DC \ and\ ED\perp BC.$
Circumcenter and Circumcircle	The point of intersection of the perpendicular bisectors of the sides of the triangle is called Circumcenter. Here $O$ is the circumcenter.
Exterior Angle and Interior Opposite Angles of a Triangle	$\angle ACD$ is the exterior angle and $\angle CBA$ and $\angle BAC$ are opposite interior angles of this exterior angle.