A right angle in a triangle has a value of 90 degrees. A right-angle triangle is a triangle whose one angle is a right angle and it is 90 degrees. The relation which is between the angles and the sides of a right triangle is said to be the basis for trigonometry’s topic.

- The side which is opposite the right angle is known as the hypotenuse.
- The sides which are just adjacent to the right angle are known as the legs.

The very famous Greek philosopher Sir Pythagoras had derived an important formula for a **right-angled triangle**. The formula of the right-angled triangle states that in a right-angled triangle, the hypotenuse square is equal to the sum of the squares of both legs. This was named after him as the theorem of Pythagoras or the Pythagorean theorem.

The formula of a right-angled triangle can be denoted in the following way:

- The hypotenuse square is said to be equal to the square sum of the base as well as the altitude.
- For a right triangle we can denote it as: (Hypotenuse)
^{2}= (Base)^{2}+ (Altitude)^{2}

**Definition of Right Angled Triangle**

A triangle that is right-angled is a triangle with one of the angles as 90 degrees as already discussed. An angle of 90-degree is known as the right angle and hence the triangle which is having a right angle is known as the right-angle triangle. In this particular triangle, the relationship which is between the various sides can be easily understood with the help of the rule of Pythagoras. The side which is opposite to the right angle in a right angle triangle is the largest side and is known as the hypotenuse.

Further, we can understand it by taking some examples that are based on the other angle values. That is, the right triangles are generally classified as a right-angled triangle that is isosceles and along with that, it is a scalene right-angled triangle as well. The lengths of the sides which are of a right-angled triangle such suppose 3, 4, 5 are known as Pythagorean triples.

**Pythagoras Theorem Formula**

The name is itself derived from the very early theorem, the Pythagorean theorem. This theorem is stating that every right triangle has a side and along with that length as well which is satisfying the formula that is: a^{2} + b^{2} = c^{2}. Thus the triplets of Pythagorean describe the 3 side length of the integer of a right triangle. However, we can say that the right triangles with sides that are non-integer do not form triplets of Pythagorean.

The relation which is established between the two can be understood by the following two theorems mentioned below:

The right angles triangle:

- In a right triangle, we can write the formula as (Hypotenuse)
^{2}= (Base)^{2}+ (Altitude)^{2}. And in the - Pythagoras theorem, we have: (Hypotenuse)
^{2}= (Perpendicular)^{2}+ (Base)^{2}.

**The Pythagorean Theorem**

The Pythagorean Theorem, which is also written as a^{2 }+ b^{2 }= c^{2} can be used to find the length of any side of a right angles triangle.The Pythagorean Theorem is one of the most famous fundamental theorems in mathematics. Thus it defines the relationship in a right-angled triangle, between all three sides. Now we are already aware of the definition of a right. We can revise it as: it is the triangle that has one of its angles as a right angle (90 degrees). The side of the triangle that is opposite to the 90-degree angle is called the hypotenuse.

The right-angled triangles have the other two sides adjacent to the right angle and these are known as the legs of the triangle.

The theorem of sir Pythagoras is thus known as the Pythagorean theorem which states that the square of the length of the hypotenuse is equal to the sum of squares of the lengths of the other two sides which are there in the right triangle. Or in even simpler words we can say that it is equal to the sum of the squares of the two legs of a right triangle to the square of its hypotenuse.

Now for instance let us call one of the legs on which the triangle rests as the base. The side which is exactly opposite to the right angle is known as its hypotenuse as already discussed in the above article. The remaining side is now known as the perpendicular. So in mathematical terms, we represent the **Pythagoras theorem formula** as (Hypotenuse)^{2} = (Perpendicular)^{2} + (Base)^{2}.

Theorems and their derivation are important in Mathematics and can be understood in a logical and easy manner at Cuemath which is a live maths tutoring platform.