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Properties of triangles.
There are a lot of formulas and techniques to find the area of a triangle. We can use many different formulas to calculate area of a triangle according to the given conditions. Here we shall derive some of the main formulas used to calculate area of a triangle.
Formulas for Area of a Triangle:
The area of a triangle is denoted by the symbol delta ( )
We shall appeal to the formula:
And the half
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Half Angle formulas?:
The Half angle formulas are stated below:
If ABC is a triangle , A , B and C are the three angles of the triangle and a , b , c are the sides opposite to the corresponding angles and
“s” is the semi perimeter or , , Then:
Proof of Half angle formula:
First of all let’s prove the half angle formula for
Using the cosine law:
Now usin
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Projection Law:
Projection law states that in any triangle:
Where , A , B , C are the three angled of the triangle and a , b , c are the corresponding opposite side of the angles.
Projection law or the formula of projection law express the algebraic sum of the projection of any two side in term of the third side.
Proof of Projection law:
To prove the projection law we shall take the
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Sine Law:
Sine law states that in any triangle ABC:
And also with some mathematics we can also prove the following :
Which is also closely related to the sine law.
Where , a , b and c are sides of a triangle , A , B and C are angles opposite to sides a , b, and c correspondingly and R is the circum-radius of the triangle as shown in following figure:
sine law
Proof of Sine
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Cosine Law states that in any triangle following equations can be implied:
Where a is the side opposite to vertex A , b is the side opposite to vertex B and c is the side oposite to vertex C.
We can prove the above stated equations by following way:
To prove the first of these formulas , we place the triangle ABC in the standard position with the vertex A at origin and the side AB along the posit
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