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Pythagorean Theorem formula is one of the most important and fundamental theorems in mathematics. It defines a relationship between the three sides of a right-angled triangle.

## This blog will explain the Pythagorean Theorem Formula with examples and derivation

The formulation and properties of a right-angled triangle is the main application of Pythagoras theorem. A right-angled triangle has one of its angles as a right angle, it means that the measurement of one of its angles is 90 degrees. The side which is opposite to the 90-degree angle is termed the hypotenuse. The other two sides which are adjacent to the right angle are called the legs of the triangle, known as perpendicular and base. The perpendicular is also known as altitude or height of the triangle.

The Pythagorean Theorem is a very
useful formula for calculating the value of the sides of a right-angled triangle.
This formula has many direct and indirect applications in the geometrical section
of **mathematics** along with its derivations and applications. In the
triangle, the hypotenuse has the longest side measurement.

It can be easily identify the longest side by looking across from the right angle triangle. The other two legs will be base and perpendicular, which are making a complete 90-degree angle. Thus the other two angles of the triangle are acute, and are called acute angles. There is no specific rule to consider the side as base or perpendicular. It does not matter at all.

The other term of Pythagoras theorem is the Pythagorean Theorem. This theorem states that the sum of the individual square of base and perpendicular of a triangle is equal to the square of the length of the hypotenuse. In other words, the square root of the sum of the squares of the two legs of a right triangle will be equal to its hypotenuse.

## So, mathematically, the presentation of the Pythagoras theorem:

Hypotenuse^{2}=Perpendicular^{2}+Base^{2}

Hypotenuse = Square root of (Perpendicular^{2}+Base^{2} )

The Pythagoras theorem was first used by Greek mathematician Pythagoras of Samos. This Greek Philosopher, along with a group of other mathematicians, formed the theorem, hence known as Pythagoras theorem. Derivation of Pythagorean Theorem formula was such that:

For a right-angled triangle ΔABC,

Side AB is termed as base, BC is termed as altitude and AC is termed as Hypotenuse.

angle sign CAB=Acute angle

angle sign BCA=Acute angle

angle sign ABC=90° (right angle)

AC^{2} = AB^{2} + BC^{2}

Right triangles always follow the rule of the Pythagoras theorem and they are called Pythagoras theorem triangles. The three sides of such a triangle are jointly called Pythagoras triples.

## How to determine whether a triangle is right-angled triangle?

If the length of three sides of a triangle are provided, then to find whether the triangle is a right-angled triangle or not, it is necessary to use the Pythagorean theorem.

The measurement of the sides of a triangle are 10, 24, and 26.

Here from the measurement, 26 is the longest side given.

It also satisfies the condition, 10 + 24 > 26

We know from the Pythagoras theorem,

AC^{2} = AB^{2} + BC^{2}
………………………(1)

So, assume AB = 10, BC = 24 and AC = 26

First we will solve R.H.S. of equation 1.

AB^{2} + BC^{2} = 10^{2} +
24^{2} = 100 + 576 = 676

Now, taking L.H.S,

AC^{2} = 26^{2} =
676

Therefore it is proved,

LHS = RHS

Therefore, the given triangle is a right-angled triangle.

There are many applications of the Pythagoras theorem, and these can be seen in our day-to-day life. Applications of the Pythagoras theorem in various field:

**Science
and Engineering for Construction fields:**

Most architects in the construction use the technique of the Pythagorean theorem to calculate the unknown dimensions. When length and altitude are known it is very easy to calculate the diameter of a particular sector. It is mainly used in two dimension structures in engineering fields.

**Another
application is for Face recognition in security cameras:**

The face recognition feature in security cameras uses the concept of the Pythagorean theorem, in such a way that the distance between the placed security camera and the location of the person is noted and the image is well-projected through the lens using the concept.

**Carpentry
and interior furniture designing:**

The
Pythagoras concept is applied in interior **furniture** designing
and the architecture of a houses and buildings construction like inclination,
steepness.

**Navigation
system:**

Sailors use this mathematical technique while traveling in the sea to find the shortest distance and route to proceed to their destination.

The calculation of diagonal connecting two buildings can be manipulated using this formula.

This formula can also be used along with trigonometry concept, to find the angle of elevation when a person looking at an object placed at the tip of a building or at the top of a pole with respect to the horizontal plane. The angle of depression is measured when the object is kept below the line of sight of the person.

**Conclusio****n:** After
reading this blog I hope you have got an idea of applying Pythagoras Theorem
Formula in mathematics as well as in real life computation. If you have any
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