The other name of Pythagoras Theorem is ‘Pythagorean Theorem’. This theorem studies the sides of the right-angled theorem. Do you know where Pythagoras theorem is used? In this context, we will learn about the Theorem in detail. In a right-angled triangle, if you are provided with only 2 sides, and you need to find out the third side, then this theorem is applied. With the help of this theorem, a right-angled triangle’s base, perpendicular, and the hypotenuse are estimated. H^{2} = P^{2} + B^{2}, where H, P, and B are hypotenuse, perpendicular, and base of a right-angled triangle respectively.

## What Does the Pythagoras Theorem State?

In a Pythagoras theorem following has been stated –

“In a structure that is a right-angled triangle, the square which makes the hypotenuse side is equal to the sum of the squares of the other two sides of the right-angled triangle”. So, from this statement, we understand that hypotenuse is the longest side, while the opposite angle measures always 90 degrees. The Pythagorean Theorem is named after a Greek Mathematician named Pythagoras.

For representing the formula, let us take the sides of the triangle for better understanding.

Suppose ‘pqr’ makes a right-angled triangle.

‘p’ – perpendicular of the right-angled triangle.

‘q’ is the base of the right-angled triangle.

‘r’ represents the hypotenuse of the right-angled triangle.

According to the statement of the Pythagoras theorem, the formula can be derived as –

r^{2} = p^{2} + q^{2}

This means, Hypotenuse^{2} = Perpendicular^{2} + Base^{2}

**Example-Based on Pythagoras Theorem**

In the question, it is a right-angled triangle, with a base measuring as 6 cm and perpendicular as 5 cm. Find out the Hypotenuse.

Soln.

Base = 6cm

Perpendicular = 5cm

Hypotenuse = x cm (assuming it to be ‘x’ cm)

Now, putting this in Pythagoras Theorem, Hypotenuse^{2} = Base^{2} + Perpendicular^{2}

x^{2} = 6^{2 + }5^{2}

x^{2 }= 36 + 25

x^{2 }= 61

x = √61

x = 7.810

## What is a Right-Angled Triangle?

In the previous section, we were talking about Pythagoras Theorem which is being applied to a right-angled theorem. Now, in this section, we will know about a right-angled triangle in detail. A right-angled triangle is a triangle that has an interior angle of 90 degrees. In this triangle, only one interior angle measures 90 degrees. For this reason, the triangle is known as a right-angled triangle. This right-angled triangle is also used in the fundamentals of trigonometry.

## The Sides of a Right-Angled Triangle

The right-angled triangle is one of the triangles whose one of the angles is equal to 90 degrees. While the other two angles which form the base and hypotenuse together sum up to be 90 degrees. The Hypotenuse side is the longest side of the right-angled triangle. While the side opposite to the 90 degrees is the smallest side.

These three sides of a right-angled triangle are very much related to each other, their relation is displayed by – Hypotenuse^{2} = Perpendicular^{2} + Base^{2}. This says that the area of the biggest side is equal to the sum of areas of the other two sides in a right-angled triangle.

## Describe the Shape of the Right Triangle

The shape of the right-angled triangle can be easily understood by the formula which is Hypotenuse^{2} = Perpendicular^{2} + Base^{2. }

We understand that the right-angled triangle is a three-sided shape that is in the form of an enclosed configuration. One side is the longest than the other two sides.

## Chalk out the properties of the Right-Angled Triangle

Let us look at some properties of the right-angled triangle.

- One of the interior angles of a right-angled triangle is 90 degrees.
- The opposite side of the 90 degrees is called the hypotenuse.
- The hypotenuse is the longest side.
- The sum of two interior angles (except the 90 degrees angle) is equal to 90 degrees. Want to explore further? Visit Cuemath for more such learning.