Pythagoras Theorem

Pythagoras Theorem
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In the 1960s, there was a musical called Merry Andrew, about a math teacher who joins the circus. In that story, there is a song that explains the Pythagoras theorem. Pythagoras said, "I wonder how far would it be to go from here to yonder without walking around that tree?" According to Pythagoras, the square of the hypotenuse is equal to the sum of the squares of the other two sides of a triangle. While Pythagoras is a genius, he had a fair share of controversy, especially one involving throwing a man into the sea for stating that the square root of 2 was irrational. However, this isn't about Pythagoras' life, it's more about the theorem.

In this lesson, you will learn about the Pythagoras theorem, its derivations, and equations followed by solved real-world problems on the Pythagoras theorem triangle and squares.

Table of Contents 

What is Pythagoras Theorem?

The Pythagoras theorem states that if a triangle is right-angled (90 degrees), then the square of the hypotenuse is equal to the sum of the squares of the other two sides. In the given triangle ABC, we have BC2 = AB2 + AC2​​. Here, ​​​​AB is the base, AC is the altitude or the height, and BC is the hypotenuse. 

A right triangle with base, altitude and hypotenuse

Pythagoras Theorem Equation

Pythagoras theorem equation helps you to solve right-angled triangle problems, using the Pythagoras equation: c2 = a2 + b2 ('c' = hypotenuse of the right triangle whereas 'a' and 'b' are the other two legs.). Hence, any triangle with one angle equal to 90 degrees will be able to produce a Pythagoras triangle. We can use this Pythagoras equation: c2 = a2 + b2 there. 

Pythagoras Theorem Formula

The Pythagoras theorem formula states that in a right triangle ABC, the square of the hypotenuse is equal to the sum of the square of the other two legs. If AB, BC, and AC are the sides of the triangle, then: BC2 = AB2 + AC2​. While if a, b, and c are the sides of the triangle, then ​c2 = a2 + b2. In this case, we can say that AB is the base, AC is the altitude or the height, and BC is the hypotenuse. 

Pythagoras Theorem Proof

Pythagoras theorem can be proved in many ways. Some of the most common and most widely used methods are by using the algebraic method proof and using the similar triangles method to solve them. Let us have a look at both of these methods individually in order to understand the proof of this theorem. 

  • Algebraic Method Proof of Pythagoras Theorem
  • Pythagoras Theorem Proof using SimilarTriangles

Algebraic Method Proof of Pythagoras Theorem

Algebraic method proof of Pythagoras theorem will help us in deriving the proof of the Pythagoras Theorem by using the values of a, b, and c (values of the measures of the side lengths corresponding to sides BC, AC, and AB respectively). Consider four right triangles ABC where b is the base, a is the height and c is the hypotenuse. Arrange these four congruent right triangles in the given square, whose side is a + b.

Theorem Proof using Algebraic Method 

The area of the square so formed by arranging the four triangles is c2. The area of square with side (a+b) = Area of 4 triangles + Area of square with side c. This implies (a+b)2 = 4×1/2×(a×b)+c2, a2+b2+2ab=2ab+c2. Therefore, a2+b2 =c2. Hence Proved.

Pythagoras Theorem Proof using Similar Triangles

Two triangles are said to be similar if their corresponding angles are of equal measures and their corresponding sides are in the same ratio. Also, if the angles are of the same measure, then we can say by using the sine law, that the corresponding sides will also be in the same ratio. Hence, corresponding angles in similar triangles will lead us to equal ratios of side lengths.

Pythagoras theorem proof using similar triangles

In triangle ABD and triangle ACB:

  • ∠A = ∠A (common)
  • ∠ADB = ∠ABC (both are right angles)

Thus, triangle ABD and triangle ACB are equiangular, which means that they are similar by AA similarity criterion. Similarly, we can prove triangle BCD similar to triangle ACB. Since triangles ABD and ACB are similar, we have AD/AB = AB/AC. Thus we can say that AD × AC = AB2. Similarly, triangles BCD and ACB are similar. That gives us CD/BC = BC/AC. Thus, we can also say that CD × AC = BC2. Now, using both of these similarity equations, we can say that AC2 = AB2 + BC2. Hence Proved.

Pythagoras Theorem Triangles

Right triangles follow the rule of the Pythagoras theorem and they are called Pythagoras theorem triangles. The length of all the three sides are being collectively called Pythagoras triplets. For example, 3, 4, and 5 can be called as one of the sets of such triangles. There are a lot more right-angled triangles which are called Pythagoras triangles. All such triangles follow one common rule: c2 = a2 + b2

Pythagoras Theorem Squares

As per the Pythagoras theorem Hypotenuse2 = Perpendicular2 + Base2 or c2 = a2 + b2, which further justifies that the area of the square built upon the hypotenuse of a right triangle will be equal to the sum of the area of the squares built upon the other two sides. And these squares are known as Pythagoras squares.


Important Topics 

Given below is the list of topics that are closely connected to Pythagoras Theorem. These topics will also give you a glimpse of how such concepts are covered in Cuemath. 


FAQs on Pythagoras Theorem 

What is the Use of the Pythagoras Theorem? 

The Pythagoras theorem works only for right-angled triangles. When any two values are known, we can apply the theorem and calculate the other.

How do you Find the Pythagoras Theorem for a Right Triangle?

The square of the hypotenuse of a right triangle is equal to the sum of the square of the other two sides. When any two values are known, we can apply the theorem and calculate the other.

What are the Applications of the Pythagoras Theorem in Real Life?

The Pythagoras theorem helps in

  • Computing the distance between points on the plane.
  • Calculating the perimeter, the surface area, the volume of geometrical shapes, and so on.
  • In architecture and construction industries.
  • In surveys 

How do you Solve A Squared B Squared C Squared? 

A squared B squared C squared implies c2 = a2 + b2, which is the formula of the Pythagoras theorem, for figuring out the hypotenuse in a right triangle when the other two sides are given.

What is the Pythagoras Theorem in Math?

Right triangles follow the rule of the Pythagoras theorem and they are called Pythagoras theorem triangles. All such triangles will follow just one common rule: c2 = a2 + b2

Can we Apply the Pythagoras Theorem to any Triangle? 

No, you can't apply Pythagoras theorem to any triangle. It needs to be a right-angled triangle only then one can use the Pythagoras theorem and obtain the relation where the sum of two squared sides is equal to the square of the third side. 

What is the Converse of Pythagoras theorem?

The converse of Pythagoras theorem is: If the sum of the squares of any two sides of a triangle is equal to the square to the third (largest) side, then it is said to be a right-angled triangle. 


Pythagoras Theorem Solved Examples 

Example 1: Kate, Jack, and Noah were having a party at Kate's house. After the party gets over, both went back to their respective houses. Jack's house was 8 miles straight towards the east, from Kate's house. Noah's house was 6 miles straight south from Kate's house. How far away were their houses (Jack's and Noah's)?

Solution: 

We can visualize this scenario as a right-angled triangle. That means Jack and Noah are hypotenuses apart from each other. So the distance between Jack's house and Noah's house can be found by: d2 = 82 + 62 = 100, where d is the distance of the house. Upon solving, we get d = 10.

Therefore, the houses are 10 miles away from each other. 

Example 2:  Julie wanted to wash her building window which is 12 feet off the ground. She has a ladder that is 13 feet long. How far should she place the base of the ladder away from the building?

Solution: 

We can visualize this scenario as a right triangle. We need to find the base of the right triangle formed. We know that, Hypotenuse2 = Base2 + Height2. Thus we can say that b2 = 132 - 122 where b is the distance of the base of the ladder from the feet of the wall of the building. Hence, we get b = 5.

Therefore, the base of the ladder is 5 feet away from the building. 


Practice Questions on Pythagoras Theorem

Here are a few Pythagoras theorem questions for you to practice. Select/Type your answer and click the "Check Answer" button to see the result. 

 
 
 
 
 
 
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