Properties of Triangles

In trigonometry we will discuss about the different properties of triangles.

We know any triangle has six parts, the three sides and the three angles are generally called the elements of the triangle.

Oblique triangle: A triangle which does not contain right angles is called an oblique triangle.

These six parts are not independent of each other i.e., various relations exist among the six parts.

We will learn how to deduce the relations between the sides and trigonometric ratios of the angles of a triangle.

Let us discuss about some of the properties of triangles.

Properties of Triangles

We know in any triangle ABC, the measures of the angles ∠ABC, ∠BCA and ∠CAB at the vertices B, C and A are denoted by the letters B, C and A respectively. The measures of the sides AB, BC and CA opposite to angles C, A and B respectively are denoted by c, a and b. The perimeter of the triangle is denoted by 2s and semi-perimeter of the triangles denoted by (a + b + c)/2 . The area of the triangle is denoted by ∆ or S. The radius of the circum-circle of the triangle is called the circum-radius and is denoted by R. The radius of the in-circle of the triangle is called the in-radius and is denoted by r. The radius of an ex-circle of the triangle is called an ex-radius and the radii of the ex-circles opposite to the angles A, B, C are denoted by r1, r2, and r3 respectively.

The six elements of a triangle are not independent and are connected by the relations A + B + C = π, a + b > c, b + c > a and c + a > b. In addition to these relations, the elements of a triangle are connected by some trigonometric relations.

● Properties of Triangles

• The Law of Sines or The Sine Rule
• Theorem on Properties of Triangle
• Projection Formulae
• Proof of Projection Formulae
• The Law of Cosines or The Cosine Rule
  
• Area of a Triangle
• Law of Tangents
• Properties of Triangle Formulae
• Problems on Properties of Triangle

11 and 12 Grade Math

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