What is a triangle?
A triangle is a closed figure formed when three line segments join end to end.
The three dots in the figure represent the points where each line segment intersects another line segment.
A triangle can be of various shapes; some of them are represented below:
Components of a triangle:
A triangle consists of the following three components:
- Sides
- Vertices
- Angles
Consider the following triangle:
The intersection points of the three line segments, or the corner points of the triangle, are called its vertices. The three vertices of the triangle are \(A\), \(B\) and \(C\).
The three line segments that join the pairs of vertices of the triangle are called its sides. The three sides of the triangle are given by \(AB\), \(BC\) and \(AC\).
The sides that intersect at a corner form an angle. The three angles formed by the triangle are \(\angle ABC\), \(\angle BAC\) and \(\angle ACB\). This can also be denoted as \(\angle B\), \(\angle A\) and \(\angle C\).
Mathematical representation of a triangle:
A triangle is named using its vertices, and the symbol used to represent a triangle is \(\Delta\).
Here, the above-given triangle is mathematically represented as \(\Delta ABC\).
The naming of triangles based on sides
Type 1: Equilateral triangle
A triangle with all three sides of equal measure is an equilateral triangle.
Here, \(\overline{AB} = \overline{BC} = \overline{AC}\).
In other words, all sides of triangle \(ABC\) are equal.
Type 2: Isosceles triangle
A triangle with two sides of equal measure and a third unequal side is an isosceles triangle.
Here, \(\overline{AB} = \overline{AC}\).
Two sides of the triangle \(ABC\) are equal while the third side is unequal.
Type 3: Scalene triangle
A triangle with all three sides of unequal measure is a scalene triangle.
Here, the three sides \(AB\), \(BC\) and \(AC\) are unequal.
The naming of triangles based on angles:
- Type - 1: Acute angled triangle
If all three angles of the triangle are acute (less than \(90°\)), then it is called an acute-angled triangle.
Example:
Here, all the three angles are less than \(90°\). That is, \(∠A=60°, ∠B=50°\) and \(∠C=70°\).
- Type - 2: Right angled triangle
If one of the angles of a triangle is a right angle \((90°)\) and the other two are acute angles (less than \(90°\)), then it is called a right-angled triangle.
Example:
Here, one among the angles is \(90°\), and the other two are acute. That is, \(∠A=40°, ∠B=50°\) and \(∠C=90°\).
- Type - 3: Obtuse angled triangle
If one of the angles of a triangle is an obtuse angle (greater than \(90°\)), and the other two are acute angles (less than \(90°\)), then it is called an obtuse-angled triangle.
Example:
Here, one angle is greater than \(90°\) and the other two are acute. That is, \(∠A=120°, ∠B=35°\) and \(∠C=25°\).