# Pythagoras Theorem

**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.

Pythagoras theorem is basically used to find the length of an unknown side and angle of a triangle. By this theorem, we can derive base, perpendicular and hypotenuse formula. Let us learn mathematics of Pythagorean 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 triangle 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 a, b and c) which have positive integer values, when squared, are put into an equation, also called a 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,

“b” is the base,

“c” is the hypotenuse.

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”

### Example

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

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

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

x^{2} = 64+36 = 100

x = √100 = 10

Therefore, the value of x is 10.

## 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 meeting AC at D.

Proof:

We know, △ADB ~ △ABC

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

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

Also, △BDC ~△ABC

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

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 theorem 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.

### Useful For

Pythagoras theorem is useful to find the sides 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} + 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,

c^{2} = a^{2} + b^{2}

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

c^{2 }= 9+16

c^{2 }= 25

c = √25

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 of 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 this statement with the help of an example.

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

Clearly, 26 is the longest side.

It also satisfies the condition, 10 + 24 > 26

We know,

c^{2 }= a^{2 }+ b^{2} ………(1)

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

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

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

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

c^{2} = 26^{2} = 676

We can see,

LHS = RHS

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

## 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}**

Let,

Perpendicular = 12 units

Base = 5 units

Hypotenuse = 13 units {since it is the longest side measure}

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

⇒ 144 + 25 = 169

⇒ 169 = 169

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

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

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

**Solution:**Given;

Perpendicular = 15 cm

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 cm

**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

ac = 4√2.

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

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## Frequently Asked Questions – 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 be 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.