**Pythagoras Theorem** is an important topic in Maths, which explains the relation between the sides of a right-angled triangle. It is also sometimes called Pythagorean Theorem. The formula and proof of this theorem are explained here. This theorem is basically used for the right-angled triangle and by which we can derive base, perpendicular and hypotenuse formula. Let us learn this theorem in detail here.

**Table of Contents:**

## Pythagoras Theorem Statement

Pythagoras theorem states that “**In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides**“. The sides of this triangles have been named as Perpendicular, Base and Hypotenuse. Here, the **hypotenuse** is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say x, y and z) which has positive integer values, when squared are put into an equation, also called a Pythagorean triple.

## Pythagoras Theorem Formula

Consider the triangle given above:

Where “a” is the perpendicular side,

“b” is the base,

“c” is the hypotenuse side.

According to the definition, the Pythagoras Theorem formula is given as:

Hypotenuse^{2} = Perpendicular^{2} + Base^{2}
= a^{2} + b^{2} |

The side opposite to the right angle (90°) is the longest side (known as Hypotenuse) because the side opposite to the greatest angle is the longest.

Consider three squares of sides a,b,c mounted on the three sides of a triangle having the same sides as shown.

By Pythagoras Theorem –

Area of square A + Area of square B = Area of square C

## Pythagoras Theorem Proof

Given: A right-angled triangle ABC.

To Prove- AC^{2} = AB^{2} + BC^{2}

Proof: First, we have to drop a perpendicular BD onto the side AC

We know, △ADB ~ △ABC

Therefore, \(\frac{AD}{AB}=\frac{AB}{AC}\) (Condition for similarity)

Or, AB^{2 }= AD × AC ……………………………..……..(1)

Also, △BDC ~△ABC

Therefore, \(\frac{CD}{BC}=\frac{BC}{AC}\) (Condition for similarity)

Or, BC^{2}= CD × AC ……………………………………..(2)

Adding the equations (1) and (2) we get,

AB^{2 }+ BC^{2 }= AD × AC + CD × AC

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

Since, AD + CD = AC

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

Hence, the Pythagorean thoerem is proved.

**Note:** **Pythagorean theorem is only applicable to Right-Angled triangle.**

### Applications of Pythagoras Theorem

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

### Pythagorean Theorem Problems

**Problem 1: **The sides of a triangle are 5,12 & 13 units. Check if it has a right angle or not.

**Solution:** From Pythagoras Theorem, we have;

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

Perpendicular = 12 units

Base = 5 units

Hypotenuse = 13 units

12^{2} + 5^{2} = 13^{2}

⇒ 144 + 25 = 169

⇒ 169 = 169

L.H.S. = R.H.S.

Therefore, the angles opposite to the 13 unit side will be at a right angle.

**Problem 2: **The two sides of a right-angled as shown in the figure. Find the third side.

**Solution: **Given;

Perpendicular = 15cm

Base = b cm

Hypotenuse = 17 cm

As per the Pythagorean Theorem, we have;

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

⇒15^{2} + b^{2} = 17^{2}

⇒225 + b^{2} = 289

⇒b^{2} = 289 – 225

⇒b^{2} = 64

⇒b = √64

Therefore, b = 8

**Problem 3:** Given the side of a square to be 4 cm. Find the length of the diagonal**.**

**Solution- **Given;

Sides of a square = 4 cm

To Find- The length of diagonal ac.

Consider triangle abc (or can also be acd)

(ab)^{2} +(bc)^{2 }= (ac)^{2}

(4)^{2} +(4)^{2}= (ac)^{2}

16 + 16 = (ac)^{2}

32 = (ac)^{2}

(ac)^{2} = 32

or ac = 4√2.

Thus, the length of the diagonal is 4√2.

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## Frequently Asked Questions on Pythagoras Theorem

### What is the formula for Pythagorean Theorem?

The formula for Pythagoras, for a right-angled triangle, is given by; c^{2}=a^{2}+b^{2}

### What is the formula for hypotenuse?

The hypotenuse is the longest side of the right-angled triangle, opposite to right angle, which is adjacent to base and perpendicular. Let base, perpendicular and hypotenuse are a, b and c respectively. Then the hypotenuse formula, from the Pythagoras statement, will be;**c**** = √(a ^{2}**

**+ b**

^{2})### Can we apply the Pythagoras Theorem for any triangle?

No, this theorem is applicable only for the right-angled triangle.

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