# What are the Main Properties of the Right-Angled Triangle?

## Right Angled Triangle: Definition

Till now you have learned about different types of triangles. Now we will focus on a specific type of triangle namely right angled triangle. Any triangle that has one of its three angles equal to a right angle i.e. 90 degrees is called a right-angled triangle.

This classification of triangles forms the basis of trigonometry and the relations between the sides of such right triangles can be understood using the Pythagorean rule, popularly known as Pythagoras theorem.

## Types of Right-Angled Triangles May Have:

**Isosceles right triangles:**These triangles have their base and height perpendicular and these two sides are also equal and include the right angle between them. The third side forms the hypotenuse.**Scalene right triangle:**A scalene right triangle has all of its sides unequal in length and angles also are unequal and the angle opposite to the longest side forms the right angle (90 degrees).

## What are the Main Properties of a Right-angled Triangle?

- In a right-angled triangle, the side opposite to the right (90 degrees) angle is the largest side of the triangle and is called the hypotenuse of the right triangle.
- The other two shorter sides are adjacent to a 90-degree angle from the base and altitude (height) of the triangle and are said to be perpendicular to one another.
- One angle of such triangles is 90 degrees and the other two acute angles form the sum of 90 degrees, thus, satisfying the angle sum property theorem of triangles.
- The right triangle forms the basis of the branch of mathematics known as Trignometry.
- If the length of all sides of a right angle triangle is integers then the triangle is considered to be a Pythagorean triangle with sides being called the Pythagorean triplet.
- The length/measure of the sides of the right-angled triangle can be given by the Pythagorean relation which says, Hypotenuse^2= base^2 + height^2
- The right triangle is a case of an isosceles right triangle if the two smaller angles of the triangle are 45 degrees each.
- The perimeter of a right triangle is the sum of lengths of all sides. The perimeter of any figure is a linear value and is always measured in unit length.

Some of the examples of a right-angled triangle can be seen in our daily life when we cut the bread slices triangular in shape, or a square piece of paper is folded diagonally, etc.

## Right Triangle Formulas

A Greek philosopher Pythagoras explained the relation between the sides of a right-angle triangle in an efficient and easy way. According to the relation, the square of the hypotenuse of the right triangle equals the sum of the square of base and the square of the height of the triangle. Using this relation one can easily find the third side of the right triangle when the other two are given.

Then related to this we have the concept of Pythagorean triplets that explains that any three numbers that satisfy the above equation of Pythagoras theorem are called Pythagorean triples. Some examples of Pythagorean triplets are: 6, 8, 10; 3, 4, 5; 12, 5, 13.

## Area of Right Angled Triangle

The right triangular area is the space or region occupied by the figure. Actually, it is the space inside it. When a ladder is kept leaning over a wall then the shadow of the ladder formed on the floor is the best example to understand the concept of the area of a right triangle.

Hence the area of right triangle, in general, is given by:

### Area of Right Angled Triangle = ½ * Base * Height

Where the two sides other than hypotenuse can be used interchangeably as base and altitude. The area is always expressed in square units. The derivation of the formula for the area of a right-angled triangle is well explained in detail at Cuemath’s website. The branch trigonometry completely arises out of the concept of right triangles.