# Triangles and their Properties

A triangle is a figure enclosed by three sides. In the figure given below ABC is a triangle with sides AB, BC and CA measuring c, a and b units respectively. Line AD represents the height of the triangle corresponding to the side BC and is denoted by h.

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In any triangle ABC

Area Ã— BC Ã— AD

*a*Ã—*h*

# Properties of a Triangle

- The sum of all the angles of a triangle = 180Â°
- The sum of lengths of the two sides > Length of the third side
- The difference of any two sides of any triangle < length of the third side
- The area of any triangle can be found by several methods:
- Area of any triangle Ã— base Ã— perpendicular to base from the opposite vertex.
- Area of any triangle where S is the semi-perimeter of the triangle and a, b and c are the sides of a triangle.
- Area of any triangle Ã— bc Sin A

Besides, there are some formulae which we use exÂclusively in some particular cases.

Example

What is the number of distinct triangles with integral valued sides and perimeter as 14?

- 6
- 5
- 4
- 3

Solution

The sum of the lengths of the two sides > the length of the third side
So, the maximum length of any particular side can be 6 units.
Now if a = 6, then b + c = 8, so the possible sets are

(6, 6, 2), (6, 5, 3) and (6, 4, 4)

If a = 5, then b + c = 9, so the possible set is (5, 5, 4)
So, the number of distinct triangles = 4

(6, 6, 2), (6, 5, 3) and (6, 4, 4)

If a = 5, then b + c = 9, so the possible set is (5, 5, 4)