**Pythagoras Theorem:** Pythagoras’ Theorem is regarded as an important concept in Maths that finds immense applications in our day-to-day life. The theorem is named after the Greek mathematician and philosopher Pythagoras which explains the relationship between the three sides of a right-angled triangle. In this article, we will be providing you with all the necessary information about Pythagoras’ Theorem – statement, explanation, formula, proof, and examples.

**DOWNLOAD FREE NCERT BOOKS FOR ALL CLASSES HERE**

## Pythagoras Theorem: Everything You Need To Know

If one of the three angles of a triangle measures 90°, then we call it a right-angled triangle. The side of the triangle opposite to the right angle is called the hypotenuse of the triangle whereas the other two sides are called base and height respectively. Pythagoras’ Theorem explains the relationship between the hypotenuse, the base, and the height of a right-angled triangle.

### Pythagoras Theorem Statement

The 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.”

This means that the area of the square on the hypotenuse of a right-angled triangle is equal to the sum of areas of the squares on the other two sides of the triangle.

### Pythagoras’ Theorem Explanation

Check out the image below:

△ABC is a right-angled triangle with ∠B as the right angle.

AB is the height or altitude of the triangle.

Length of AB = *a*.

BC is the base of the triangle.

Length of BC = *b*.

AC, the side opposite to the right angle is the hypotenuse of the triangle.

Length of AC = *c*.

Let us now imagine three squares on the three sides of the triangle.

Now, as per the theorem, the area of the square on the hypotenuse of the triangle is equal to the sum of the areas of the squares on the other two sides.

This means:

### Pythagoras Theorem Formula

Referring to the above image, the theorem can be expressed as:

*(Hypotenuse) ^{2} = (Height)^{2} + (Base)^{2} or *

*c*^{2}=

*a*^{2}

*+ b*^{2}

### Pythagoras Theorem Proof

The proof of Pythagorean Theorem is provided below:

Let us consider the right-angled triangle △ABC wherein ∠B is the right angle (refer to image 1).

In order to prove (AB)^{2} + (BC)^{2} = (AC)^{2}, let’s draw a perpendicular line from the vertex B (bearing the right angle) to the side opposite to it, AC (the hypotenuse), i.e. BO ⊥ AC.

Now, in △ABC and △ABO, we have:

- i) ∠A is common.
- ii) ∠AOB = ∠ABC (Both are 90°)

**Therefore, △ABC ~ △ABO (By AA-similarity)**

So, AO/AB = AB/AC.

=> **(AB)**^{2}** = AO × AC** ——– (1)

Now, in △ABC and △OBC, we have:

- i) ∠C is common.
- ii) ∠BOC = ∠ABC (Both are 90°)

**Therefore, △ABC ~ △OBC (By AA-similarity)**

So, OC/BC = BC/AC.

=> **(BC)**^{2}** = OC × AC** ——– (2)

Adding equation 1 and equation 2, we have:

(AB)^{2} + (BC)^{2} = AO × AC + OC × AC

=> (AB)^{2} + (BC)^{2} = AC (AO + OC)

=> (AB)^{2} + (BC)^{2} = AC × AC (Now, since AO + OC = AC)

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

Hence, the Pythagorean Theorem is proved.

### Applications Of Pythagorean Theorem

Apart from solving various mathematical problems, Pythagorean Theorem finds applications in our day-to-day life as well, such as, in:

- i) Architecture and construction, let’s say to construct a square corner between two walls, to construct roofs, etc.
- ii) Cartography
- iii) Surveying
- iv) Painting and design
- v) Navigation

### Examples Of Pythagoras Theorem

Some example problems related to Pythagorean Theorem are as under:

*Example 1: The length of the base and the hypotenuse of a triangle are 6 units and 10 units respectively. Find the length of the third side (height).*

**Solution:**

Length of base = 6 units

Length of hypotenuse = 10 units

We know that ** (Hypotenuse)^{2} = (Height)^{2} + (Base)^{2}** .

=> (10)

^{2}= (Height)

^{2}+ (6)

^{2}

=> 100 = (Height)

^{2}+ 36

=> (Height)

^{2}= 100 – 36

=>

**(Height)**

^{2}

**= 64**

Therefore, Height = √(64) = 8 units.

*Example 2: The length and breadth of a rectangle are 5 units and 11 units respectively. Find the length of the diagonal.*

**Solutions:**

Each of the four angles of a rectangle measures 90°.

Therefore, each diagonal divides the rectangle into two right-angled triangles, with the diagonal being the hypotenuse of each of the triangles and the length and breadth being the other two sides.

As** (Hypotenuse)^{2} = (Height)^{2} + (Base)^{2}**,

**= 25 + 121 = 146**

*(Hypotenuse)*^{2}= (5)^{2}+ (11)^{2}Therefore, Hypotenuse (Diagonal of the Rectangle) = √(146) = 12.083 units.

So, now you know everything about Pythagoras Theorem. Solve more questions of varying types and master the concept.

### Pythagoras Theorem: Important FAQs

Some of the important FAQs related to the Pythagoras Theorem are:

**Q1: What is Pythagorean theorem?**

**Ans:** Pythagoras Theorem can be stated as “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.”

**Q2: Who invented the Pythagorean theorem?**

**Ans:** Pythagorean theorem is named after a Greek mathematician Pythagoras.

**Q3: How can I prove the Pythagorean Theorem?**

**Ans:** You can prove the Pythagorean Theorem in three ways:

– Using Coordinate Geometry

– Using Trigonometry

– Using Similarity

This article contains the proof of the Pythagorean Theorem from the triangle similarity method.

**Q4: What are some real life uses of the Pythagorean theorem?**

**Ans:** Pythagoras Theorem has a lot of real life uses. Construction workers, Architects, Carpenters, Framers, etc. all use its concepts.

**DOWNLOAD NCERT SOLUTIONS FROM CLASS 6 TO 12 HERE**

### Important Study Resources Of Maths Available At Embibe!

Mathematics is an important subject not only for your school-level exam but also for almost every competitive exam such as JEE Main, CAT, and many government job recruitment exams. If you want to pursue Engineering, then its concepts will follow you in one way or the other. It’s for the best that you strengthen your knowledge base from the foundation concepts.

Some other helpful articles by Embibe are provided below:

Not limited to these formulas, Embibe gives you a platform to practice K-12 questions free of cost. Make the best use of these study materials and master the subject. Find the below Maths practice questions for Class 8, 9, 10, 11, and 12. The links are provided below:

a. Class 8 Maths Practice Questions |

b. Class 9 Maths Practice Questions |

c. Class 10 Maths Practice Questions |

d. Class 11-12 Maths Practice Questions |

**ATTEMPT FREE JEE MAIN MOCK TEST SERIES HERE**

*We hope this article on Pythagoras Theorem has provided significant value to your knowledge. If you have any queries or suggestions, feel free to write them down in the comment section below. We will love to hear from you. Embibe wishes you all the best of luck!*

**334**Views