**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 the Pythagorean Theorem. The formula and proof of this theorem are explained here with examples. This theorem is basically used for the right-angled triangle andÂ 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,Â the square of the hypotenuse side is equal to the sum of squares of the other two sides**“.Â The sides of this triangle have been named as Perpendicular, Base and Hypotenuse. Here, the **hypotenuse** is the longest side, as it is opposite to the angle 90Â°. The sides of a right triangle (say x, y and z) which have positive integer values, when squared, are put into an equation, also called aÂ Pythagorean triple.

### History

The theorem is named after a greek Mathematician called Pythagoras.

## 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}Â = Perpendicular^{2}Â + Base^{2}Â
Â = a^{2}Â + b^{2}Â Â |

The side opposite to the right angle (90Â°)Â Â 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

### Example

The examples of theorem based on the statement given for right triangles is given below:

ConsiderÂ a right triangle, given below:

Find the value of x.

X is the side opposite to right angle, hence it is a hypotenuse.

Now, by the theorem we know;

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

x^{2} = 8^{2} + 6^{2}

x^{2} = 64+36 = 100

x = âˆš100 = 10

Therefore, we found the value of hypotenuse here.

## Pythagoras Theorem Proof

Given: A right-angled triangle ABC, right-angled at B.

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

Construction: Draw a perpendicular BD joining AC at D.

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

We know, â–³ADB ~ â–³ABC

Therefore, \(\frac{AD}{AB}=\frac{AB}{AC}\) (corresponding sides of similar triangles)

Or, AB^{2Â }= ADÂ Ã— AC ……………………………..â€¦â€¦..(1)

Also, â–³BDC ~â–³ABC

Therefore, \(\frac{CD}{BC}=\frac{BC}{AC}\) (corresponding sides of similar triangles)

Or, BC^{2}= CD Ã— AC ………………………………â€¦â€¦..(2)

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

AB^{2Â }+ BC^{2Â }= ADÂ Ã— AC +Â CD Ã— AC

AB^{2Â }+ BC^{2Â }= AC (AD + CD)

Since, AD + CD = AC

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

Hence, the Pythagorean theorem is proved.

**Note:**Â **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.

### Useful For

Pythagoras theorem is useful to find the sides and angles of a right-angled triangle. If we know the two sides of a right triangle, then we can find the third side.

### How to use?

To use this theorem, remember the formula given below:

c^{2} = a^{2}Â + b^{2}

Where a, b and c are the sides of the right triangle.

For example, if the value of a = 3 cm, b = 4 cm, then find the value of c.

We know,Â

c^{2} = a^{2}Â + b^{2}

c^{2} = 3^{2}+4^{2}

c^{2Â }= 9+16

c^{2Â }= 25

c=âˆš25

c=5

Hence, the third side is 5 cm.

As we can see, a + b > c

3 + 4 > 5

7 > 5

Hence, c = 5 cm is the hypotenuse of the given triangle.

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

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

Let us understand with the help of an example.

Suppose a triangle with sides 10, 24, and 26 are given.Â

Clearly, 26 is the longest side.

It also satisfy the condition, 10+24>26

We know,

c^{2Â }= a^{2Â }+ b^{2}Â Â â€¦â€¦â€¦(1)

So, let a=10,b=24 and c=26

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

a^{2}Â + b^{2} = 10^{2}Â + 24^{2}Â = 100+576 = 676

Now, taking L.H.S, we get;

C^{2} = 26^{2}Â = 676

We can see,Â

LHS = RHS

Therefore, the given triangle is a right triangle, as it satisfies the theorem.

## Pythagorean Theorem Problems

**Problem 1:Â **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}

â‡’ 144 + 25 = 169

â‡’ 169 = 169Â

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

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

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

**Solution:Â**Given;

Perpendicular = 15cm

Base = b cm

Hypotenuse = 17 cm

As per the Pythagorean Theorem, we have;

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

â‡’15^{2} + b^{2} = 17^{2}

â‡’225 + b^{2} = 289

â‡’b^{2} = 289 – 225

â‡’b^{2} = 64

â‡’b = âˆš64

Therefore,Â b = 8 cm

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

**Solution-Â **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)^{2Â }= (ac)^{2}

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

16 + 16 = (ac)^{2}

32 = (ac)^{2}

(ac)^{2} = 32

or ac = 4âˆš2.

Thus, the length of the diagonal isÂ 4âˆš2.

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## Frequently Asked Questions â€“ FAQs

### 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****Â = âˆš(a ^{2}**

**Â + 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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