# Pythagorean Theorem

The Pythagorean Theorem, also known as Pythagoras theorem is a mathematical relation between the 3 sides of a right triangle, a triangle in which one of 3 angles is 90°. It was discovered and named after the Greek philosopher and mathematician of Samos, Pythagoras.

Does Pythagorean Theorem Work on All Triangles

No, the Pythagorean Theorem works only for right triangles. Thus, it helps to test whether a triangle is right triangle or not.  The theorem is also used to find the length of one side of a right triangle when the other two sides are known.

## History

The theorem was first evident on a 4000-year-old Babylonian tablet (beginning about 1900 B.C.) now known as Plimpton 322. Much later in 570–500/490 BCE, the relationship was popularized when Pythagoras stated it explicitly.

## Theorem

When a triangle has a right angle (90°) and squares are made on each of the 3 sides, then the biggest square has an area equal to the sum of the areas of the other 2 squares.

Thus, the Pythagorean Theorem states that the area of the square formed by the longest side of the right triangle (the hypotenuse) is equal to the sum of the area of the squares formed by the other two sides.

### Derivation

In the above figure,

Area of □ A = a2

Area of □ B = b2

Area of □ C = c2

Now, according to the Pythagorean Theorem,

Area of □ A + Area of □ B = Area of □ C

(a × a) + (b × b) = (c × c)

a2 + b2 = c2

The above relation is useful to find an unknown side of a right triangle when the lengths of the other 2 sides are known.

## Formula

The equation that represents the Pythagorean Theorem in mathematical form is given below:

Let us find out how it works using an example.

Consider a right triangle with side lengths 3, 4, 5.

In the above figure, let us use the Pythagorean Theorem

32 + 42 = 52

=> 9 + 16 =25

9, 16, & 25 are the areas of the three squares

This proves how the Pythagorean Theorem works

Let us solve an example to understand the concept better

Find the hypotenuse of a right-angle triangle with the other 2 sides 15 cm and 20 cm.

Solution:

As we know,
In a right angle triangle,
a2 + b2 = c2, here c = length of the hypotenuse, a = 15 cm, c = 20 cm
=> 152 + 202 = c2
=> 225 + 400 = c2
=> c2 = 625
=> c = 25 cm
Thus, the hypotenuse of a right triangle with sides 15 cm and 20 cm is 25 cm

Find b in the given right angle triangle.

Solution:

As we know,
In a right angle triangle,
a2 + b2 = c2, here c = 5, a = 3 cm
=> 32 + b2 = 52
=> b2 = 52 – 32
=> b2 = 25 – 9
=> b2 = 16
=> b = 4 cm

Find whether it is a right angle triangle.

Solution:

As we know,
In a right angle triangle,
a2 + b2 = c2, here a = 9, b = 12, c = 15 cm
=> 92 + 122
=> 81 + 144
=> 225
As, 152 = 225
Thus, the above relation holds true for the Pythagorean Theorem, thus it is a right angle triangle.

Does an 8, 15, 16 triangle have a Right Angle?

Solution:

As we know,
In a right angle triangle,
a2 + b2 = c2
Here,
=> 82 + 152
=> 64 + 225
=> 289
But, 162 = 256
Thus, the above relation does not hold true for the Pythagorean Theorem, thus it is not a right-angle triangle.

## Proofs

The Pythagorean Theorem can be proved in many ways. The 2 most common ways of proving the theorem are described below:

### Algebraic Method

This method helps us to prove the Pythagorean Theorem by using the side lengths.

Let us consider 4 right triangles with side lengths a, b, & c, where c is the length of the hypotenuse and ‘a’ and ‘b’ are the lengths of the other 2 sides

If we arrange the 4 right triangles in a square of length (a + b), we can derive the equation of the Pythagorean Theorem as shown below:

Area of □ EFGH = c × c = c2

Again,

Area of □ ABCD = Area of □ EFGH + Area of 4 right △s

=> (a + b)2 = c2 + 4(1/2 × b × a)

=> (a + b)2 = c2 + 2ab

=> a2 + b2 + 2ab = c2 + 2ab

=> a2 + b2 = c2

This proves the Pythagorean Theorem

### Using Similar Triangles

As we know,

∴ CD/CB = CB/CA (Corresponding Sides of Similar Triangles are Equal)

=> CB2 = CD × CA …….. (1)

Similarly,

△BDA ~ △CBA

∴ AD/BA = BA/CA (Corresponding Sides of Similar Triangles are Equal)

=> BA2 = AD × CA …….. (2)

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

CB2 + BA2 = (CD × CA) + (AD × CA)

=> CB2 + BA2 = CA (CD + AD)

Since, CD + AD = CA

∴ CA2 = CB2 + BA2

This proves the Pythagorean Theorem

## Applications: What is It Used For

Through learning the basic concepts of the Pythagorean Theorem is important to determine whether a triangle is a right triangle or not. But, we are even more curious in understanding the applications of the Pythagorean Theorem.

Some common real-life applications of the Pythagorean Theorem are given below:

1. Two-dimensional navigation where it is used for calculate the distance between the ship and the point of navigation. The same principles are also used for air navigation. For example, a plane can use its height above the ground and its distance from the source to its destination.
2. Cartographers use to calculate and survey the numerical distances and heights between different points before a map is drawn. It is also used to calculate the steepness of slopes of hills or mountains.
3. Given 2 straight lines, it is used for calculating the length of the diagonal line connecting them. This application is frequently used in architecture, woodworks, or other physical construction projects. It is also used for laying out square angles for making design in buildings.
4. In oceanography, it is used for calculating speed of sound waves in water.
5. Making of tv screens, computer screens, and solar panels
6. Works to determine the height of ladder required to painta wall of certain height
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