A very common and important topic in Mathematics, Pythagoras Theorem, or the Pythagorean Theorem. It explains the relation between the different sides of a right-angled triangle. **The theorem states that â€śIn a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides of the triangle.â€ť **

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The other two sides here are the Perpendicular and the Base of the triangle. Therefore, in a right-angled triangle, the Hypotenuse becomes its longest side, since it is opposite to the 90Â° angle.

**FORMULA OF PYTHAGOREAN THEOREM**

In a right-angled triangle,

A is the Perpendicular

B is the Base

C is the Hypotenuse

*Therefore, according to the definition of the Pythagorean Theorem, the formula would be:*

**HypotenuseÂ˛= PerpendicularÂ˛ + BaseÂ˛**

In other words, it would be:

**CÂ˛= AÂ˛+ BÂ˛**

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**PROOF OF PYTHAGORAS THEOREM:**

In a right-angled triangle, the base and the perpendicular together make up an angle of 90 degrees with each other. therefore, according to the Pythagorean Theorem, â€śthe square of the hypotenuse is equal to the sum of a base square and perpendicular square.â€ť

In order to prove this theorem,

Let us suppose a triangle ABC, which is right-angled at B.

**We have to prove that: ACÂ˛= ABÂ˛ + BCÂ˛**

**To explain:** We draw a perpendicular line BD which meets AC at D.

**Proof:**

We know by the theorem that if a perpendicular is drawn from the vertex of the right angle to the hypotenuse of a right-angled triangle, then the two triangles on both the sides of the perpendicular are similar to each other as well as to the whole triangle.

Therefore,

â–łADB ~ â–łABC

Hence,

AD/AB = AB/AC (Condition for similarity)

Or, AB^{2} = AD Ă— AC (1)

Also, â–łBDC ~â–łABC (By applying the same theorem)

Therefore,

CD/BC = BC/AC (Condition for similarity)

Or,

BC^{2}= CD Ă— AC (2)

Now,

By adding the equations (1) and (2) we get,

AB^{2} + BC^{2} = AD Ă— AC + CD Ă— AC

AB^{2} + BC^{2} = AC (AD + CD)

Since, AD + CD = AC

**Therefore, AC ^{2} = AB^{2} + BC^{2}**

**Hence, the Pythagorean theorem is proved.**

**APPLICATIONS OF PYTHAGOREAN THEOREM**

Some applications of the Pythagoras Theorem are:

- To know whether the triangle is a right-angled triangle or not
- To find the diagonal of a square
- In a right-angled triangle, we can calculate the length of any side if we know the length of the other two sides

**Frequently Asked Questions- Pythagoras Theorem**

**Q. How do you find the Pythagorean Theorem?**

In a right-angled triangle, **HypotenuseÂ˛= PerpendicularÂ˛ + BaseÂ˛**

**Q. What is Pythagoras Theorem?**

It is a fundamental theorem of Mathematics that defines the relationship between three sides of a right-angled triangle.

**Q. What is the Pythagorean theorem in simple terms?**

Simple definition of Pythagorean theorem is that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides.

**Q. How is the Pythagorean Theorem useful in real life?**

This theorem is useful in architecture, construction projects, woodworking and other such projects, where you need to calculate the length of the diagonal connecting given two straight line.