Lesson Triangles and its basic properties

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In this lesson I will discuss about different types of triangle and their basic properties.
i.e. Acute Triangle, Obtuse Triangle,Equilateral Triangle, Isosceles Triangle, Scalene Triangle.

To identify the type of triangle we need to know the measure of angles and the side lengths.

<A HREF=Acute_triangle.wikipedia>Acute Triangle</A>: 
A triangle in which all three angles are acute angles (less than 90 degree). 
As shown in the diagram below. Here all the angles are less than 90 degree and hence, 
it is an Acute Triangle.
{{{drawing( 160, 160, -0.5, 6.5, -0.5, 5.0, line( 0, 0, 6, 0 ), line( 0, 0, 3, 4 ),line( 6, 0, 3, 4), locate( 0, 0, A ),locate( 3.5, 4, C ),locate( 6, 0, B ))}}}

How will you find the type of a triangle if only side lengths are given?

You should try to find answer where we have the link between angle and side lengths. 
You should look at <A HREF=Cosine_rule.wikipedia>Cosine rule</A>
{{{Cos(C)= (a^2+ b^2-c^2)/(2*a*b)}}}

So if the angle C is Acute then {{{Cos(C)>0}}} and hence from the Cosine rule: 
{{{a^2+ b^2-c^2>0= a^2+ b^2>c^2}}} as side lengths are always positive,so the 
denominator will not play any role here. This is true for each angle(A,B and C).
So we need to check {{{a^2+ b^2>c^2}}}, {{{b^2+ c^2>a^2}}} and{{{c^2 + a^2>b^2}}}simultaneously.

<A HREF=Obtuse_triangle.wikipedia>Obtuse Triangle</A>: 
A triangle in which one of the angles is an obtuse angle(more than 90 degree). 
As shown in the diagram below. Here angle {{{C}}} is more than 90 degree and hence, 
it is an Obtuse Triangle .
{{{drawing( 160, 160, -0.5, 6.5, -0.5, 5.0, line( 0, 0, 6, 0 ), line( 0, 0, 3, 1.5 ),line( 6, 0, 3, 1.5), locate( 0, 0, A ),locate( 3.2, 2, C ),locate( 6, 0, B ))}}}

Similarly here you can also find the type of triangle by Cosine Rule

{{{Cos(C)= (a^2+ b^2-c^2)/(2*a*b)}}}

So if the angle {{{C}}} is Obtuse then {{{Cos(C)<0}}} and hence from the 
Cosine rule: {{{a^2+ b^2-c^2<0= a^2+ b^2<c^2}}} as side lengths are always positive so the 
denominator will not play any role here.

<A HREF=Isosceles_triangle.wikipedia>Isosceles Triangle</A>:
A triangle in with (at least) two sides are equal.Angles opposite to equal side lengths are equal.
In the diagram below, the two equal sides have length {{{b}}} and the remaining side has length {{{c}}}. 
Hence, Angle {{{A}}} and {{{B}}} are equal.

{{{drawing( 160, 160, -0.5, 6.5, -0.5, 5.5, line( 0, 0, 6, 0 ), line( 0, 0, 3, 1.5 ),line( 6, 0, 3, 1.5), locate( 0, 0, A ),locate( 3.2, 2, C ),locate( 6, 0, B ),locate( 1.5, 1.5, b ),locate( 4.5, 1.5,b),locate( 3, 0, c ))}}}

<A HREF=Equilateral_triangle.wikipedia>Equilateral Triangle</A>.:
A triangle in which all three sides are of equal length. It is also called regular triangle.
An Equilateral Triangle is special case of an Isosceles Triangle with all three sides equal.
In the diagram below, each side is of equal length {{{a}}} and each angle is of equal measure{{{A}}}.
Hence, 
{{{A=B=C}}}
{{{A+A+A=180}}}
{{{ A=B=C=60}}}

{{{drawing( 160, 160, -0.5, 6.5, -0.5, 5.0, line( 0, 0, 6, 0 ), line( 0, 0, 3, 3.5 ),line( 6, 0, 3, 3.5), locate( 0, 0, A ),locate( 3.2, 3.8, C ),locate( 6, 0, B ),locate( 1.3, 2.3, a),locate( 4.5, 2.3,a),locate( 3, 0, a ))}}}

<A HREF=Scalene_triangle.wikipedia>Scalene Triangle</A>:
A triangle in which all three sides are of unequal lengths. As shown in the diagram below

{{{drawing( 160, 160, -0.5, 6.5, -0.5, 5.0, line( 0, 0, 6, 0 ), line( 0, 0, 1.5, 3.5 ),line( 6, 0, 1.5, 3.5), locate( 0, 0, A ),locate( 1.7, 3.8, C ),locate( 6, 0, B ),locate( 0.5, 2.4, b),locate( 4.1, 2.3,a),locate( 3, 0, c))}}}

For more information on Triangles refer to <A HREF=Triangles.wikipedia>wikipedia</A>.