What is a triangle?
A triangle is a closed figure formed when three line sements intersect each other at their endpoints.
Components of a triangle:
A triangle consists of the following three components:
- The three line segments of the triangle are called its sides.
- The intersecting points of the three line segments of the triangle are called its vertices.
- The sides intersecting at a corner gives rise to an angle.
The naming of triangles based on sides
Type 1: Equilateral triangle
A triangle with all three sides of equal measure is an equilateral triangle.
Type 2: Isosceles triangle
A triangle with two sides of equal measure and a third unequal side is an isosceles triangle.
Type 3: Scalene triangle
A triangle with all three sides of unequal measure is a scalene triangle.
Triangle Inequality:
The triangle inequality is a fundamental property of triangles in geometry. It provides a necessary and sufficient condition for three line segments to form a triangle.
Triangle Inequality for sum of two sides:
The sum of the length of any two sides of a triangle is always greater than the length of the third side.
Mathematically, for any triangle \(\Delta ABC\) with sides \(a\), \(b\) and \(c\) opposite to vertices \(A\), \(B\) and \(C\) respectively, the triangle inequality states that:
\(a + b > c\)
\(b + c > a\) and
\(c + a > b\).
Conversely, if any three positive numbers \(a\), \(b\) and \(c\) satisfy the triangle inequalities, then it is always possible to construct a triangle with sides \(a\), \(b\) and \(c\).
Triangle inequality for the difference of two sides:
In any triangle, the difference in the length of any two sides of a triangle is always lesser than the third side.
Mathematically, for any triangle \(\Delta ABC\) with sides \(a\), \(b\) and \(c\) opposite to vertices \(A\), \(B\) and \(C\) respectively, the triangle inequality states that:
\(a - b < c\)
\(b - c < a\) and
\(c - a < b\)