The Pythagorean Theorem is one of the most important fundamental theorems in math. Also, the Pythagorean theorem defines the relationship between all sides of a right-angle triangle. If you already know the definition and properties of a right-angled triangle. It is the triangle with one of its angles as a right angle, that is, 90 degrees. The hypotenuse is the side that is perpendicular to the 90-degree angle. The legs of the triangle are the other two sides that border the right angle.

The Pythagorean or Pythagoras theorem asserts that the square of the hypotenuse’s length equals the sum of the squares of the hypotenuse’s two other side lengths in a right-angled triangle. Alternatively put, the square of a right triangle’s hypotenuse equals the sum of the squares of its two legs.

Lets Identifying the base of a triangle as one of its legs will be helpful. Its hypotenuse is, as we already know, the side opposite the right angle. The remaining side is called the perpendicular. In other words, the Pythagorean theorem is represented mathematically as:

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

## Pythagorean Theorem Derivation

Let’s consider a right-angled triangle ΔABC. Here below we provide a figure whic is right-angled at B.

Let’s BD be perpendicular to the side AC.

From the above given triangles figure, consider the Triangle ΔABC and Triangle ΔADB,

In ΔABC and ΔADB,

∠ABC = ∠ADB = 90°

∠A = ∠A → common

Using the AA criterion for the similarity of triangles,

ΔABC ~ ΔADB

Therefore, AD/AB = AB/AC

⇒ AB^{2} = AC x AD ……(1)

Consideration should be given to ΔABC and ΔBDC from the diagram shown below.

∠C = ∠C → common

∠CDB = ∠ABC = 90°

Using the Angle Angle(AA), we come to a definitive conclusion regarding triangle similarity:

ΔBDC ~ ΔABC

Therefore, CD/BC = BC/AC

⇒ BC^{2} = AC x CD …..(2)

From the similarity of triangles: ∠ADB = ∠CDB = 90°

So if a perpendicular is drawn from the right-angled vertex of a right triangle to its hypotenuse, then both triangles formed on either side are similar in form as well as being in proportion with each other and the whole triangle.

**To Prove: **AC^{2} =AB^{2} +BC^{2}

By adding equation (1) and equation (2), we got:

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

AB^{2} + BC^{2} = AC (AD + CD) …..(3)

Since AD + CD = AC, substitute this value in equation (3).

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

Now, it becomes

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

**Hence, Pythagoras theorem is proved. **