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.

Acute Triangle:
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.

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 Cosine rule
Cos%28C%29=+%28a%5E2%2B+b%5E2-c%5E2%29%2F%282%2Aa%2Ab%29

Cos%28C%29=+%28a%5E2%2B+b%5E2-c%5E2%29%2F%282%2Aa%2Ab%29

So if the angle C is Acute then Cos%28C%29%3E0

and hence from the Cosine rule:
a%5E2%2B+b%5E2-c%5E2%3E0=+a%5E2%2B+b%5E2%3Ec%5E2

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%5E2%2B+b%5E2%3Ec%5E2

and

simultaneously.

Obtuse Triangle:
A triangle in which one of the angles is an obtuse angle(more than 90 degree).
As shown in the diagram below. Here angle

is more than 90 degree and hence,
it is an Obtuse Triangle .

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

Cos%28C%29=+%28a%5E2%2B+b%5E2-c%5E2%29%2F%282%2Aa%2Ab%29

So if the angle

is Obtuse then Cos%28C%29%3C0

and hence from the
Cosine rule: a%5E2%2B+b%5E2-c%5E2%3C0=+a%5E2%2B+b%5E2%3Cc%5E2

as side lengths are always positive so the
denominator will not play any role here.

Isosceles Triangle:
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

and the remaining side has length

.
Hence, Angle

and

are equal.

Equilateral Triangle.:
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

and each angle is of equal measure

.
Hence,

Scalene Triangle:
A triangle in which all three sides are of unequal lengths. As shown in the diagram below

For more information on Triangles refer to wikipedia.

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