Pythagoras Theorem Formula

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Pythagoras Equation Basics

Pythagoras theorem is one of the most popular and most important theorems that forms the basics of a separate stream of Mathematics called trigonometry. The Pythagoras theorem formula establishes a relationship between the sides of the right triangle. A right triangle is a three sided closed geometric plane figure in which one of the 3 angles is 900. In a right triangle, the side opposite to the right angle is the longest side and is called the hypotenuse. The horizontal side forming the right angle is called the base of the right triangle and the vertical side is the hypotenuse. 


What is Pythagorean Theorem?

Pythagorean theorem is a theorem in geometry that relates all the three sides of a right triangle. It states that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. 

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For example, if ΔABC is a right triangle with sides AB, BC and AC such that the triangle is right angled at B, then the Pythagoras equation is written as:


AB2 + BC2 = AC2


In the above equation, AC is the side opposite to the angle ‘B’ which is a right angle. Hence AC is the base, BC and AB are base and perpendicular respectively.

To summarize what is the Pythagorean theorem formula in general we can write that in any right triangle, 


(Hypotenuse)2 = (Base)2 + (Perpendicular)2


What is the Formula of Pythagoras Theorem Logic?

To understand the logical proof of Pythagoras Theorem formula, let us consider a right triangle with its sides measuring 3 cm, 4 cm and 5 cm respectively. Construct a perfect square on each side and divide this perfect square into unit squares as shown in figure. Now count the number of unit squares on each side of the right triangle. The number of unit squares on the sides measuring 3 cm, 4 cm and 5 cm are 9, 16 and 25 respectively. We can observe that the number of unit squares on the side measuring 5 cm is equal to the sum of the number of unit squares on the sides measuring 3 cm  and 4 cm. 

i.e. 25 = 16 + 9 

25 = 25

This activity can be tried with any right triangle and can be found that the number of unit squares on the longest side is equal to the number of unit squares on the other two sides taken together. This clearly explains the meaning of what is the formula of Pythagoras theorem.

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What is the Pythagorean Theorem Formula Proof?


Statement of Pythagoras Equation:

Pythagorean theorem states that the square on the hypotenuse is equal to the sum of the squares on the other two sides in a right triangle. 

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Data: 

Consider a right triangle ABC as shown in the figure above. 

The side AC is the hypotenuse and the angle B is 900.


To Prove:

AC2 = AB2 + BC2


Construction:

Draw BD perpendicular to AC


Proof:

Statement

Reason

ΔABC ~ ΔADB  

In a right triangle, if a line is drawn from the vertex of the right angle to the hypotenuse, then the two newly formed smaller right triangles are similar to each other and also to the larger triangle.  

[\frac{AD}{AB}\] = [\frac{AB}{AC}\]

If two triangles are similar, then their corresponding sides are proportional.

AB2 = AD . AC -- (1)


ΔABC ~ ΔBDC  

In a right triangle, if a line is drawn from the vertex of the right angle to the hypotenuse, then the two newly formed smaller right triangles are similar to each other and also to the larger triangle.  

[\frac{CD}{BC}\] = [\frac{BC}{AC}\]

If two triangles are similar, then their corresponding sides are proportional.

BC2 = CD . AC -- (2)


AB2 + BC2 = AD . AC + CD . AC 

Adding (1) and (2)

AB2 + BC2 = AC (AD + CD)

Taking AC as common

AB2 + BC2 = AC . AC

AB2 + BC2 = AC2

From the construction and diagram, 

AD + DC = AC


Worked Examples to Understand What is Pythagorean Theorem

1. If a triangle has the sides 7 cm, 8 cm and 6 cm respectively, check whether the triangle is a right triangle or not. 

Solution:

A triangle is said to be a right triangle if and only if the square of the longest side is equal to the sum of the squares of the other two sides. In the given question, the side measuring 8 cm is the longest side.

For the triangle to be a right triangle,

82 = 72  + 62

64 = 49 + 36

64 = 85

However, 64 is never equal to 85. Therefore, the given triangle is not a right triangle. 


2. Find the height of the ladder whose top touches the wall at a height 12 m above the floor and the foot of the ladder is 5 m away from the wall on the floor. 

Solution:

Considering that the floor and the wall are at right angles to each other, the line joining the bottom of the wall at the foot of the ladder forms the base of the right triangle and the line joining the tip of the ladder and the bottom of the ladder forms the perpendicular. The ladder is same as the hypotenuse of the right triangle. 


According to the Pythagoras Equation:

(Hypotenuse)2 = (Base)2 + (Perpendicular)2 

(Hypotenuse)2 = 52 + 122

(Hypotenuse)2 = 25 + 144 

(Hypotenuse)2 = 169

Hypotenuse = √169

Hypotenuse= 13 

Therefore, the height of the ladder is 13 m.


Fun Facts About Pythagoras Formula:

  • A Greek Mathematician named Pythagoras was the first person to structure this theorem and hence it is named after him. 

  • Pythagoras theorem formula is used in almost all the applications where heights and distances are to be measured. 

FAQ (Frequently Asked Questions)

1. What is the Formula of Pythagoras Theorem Practical Applications?

Pythagoras theorem finds a plethora of applications in daily life where heights and distances are to be measured. A few applications where Pythagoras formula plays a vital role in computation of heights and distances are listed below.

  • Construction and architecture.

  • Air, water and road navigation systems.

  • Constructing square angles in construction of buildings and machine parts.

2. What is Pythagorean Theorem Converse?

The converse of the Pythagorean theorem is the reverse of the statement of Pythagoras equation. The pythagoras theorem converse states that, if in any triangle, the square on one side is equal to the sum of the squares on the other two sides, then that triangle is a right triangle. This converse of Pythagoras theorem can be used to find whether any triangle whose sides are given is a right triangle or not.