**Why is it called Pythagoras Theorem? **

Pythagorean theorem, also known as Pythagoras theorem, took the name of ancient **Greek Mathematician Pythagoras **during the 500 B.C because as far the books say, he was the first person who offered a proof of the theorem.

**What is the history of Pythagoras Theorem?**

It has been argued that the **ancient Babylonians** already understood the theorem long before the invention by Pythagoras. They knew the relationship between the sides of the triangle and while solving for the hypotenuse of an isosceles triangle, they came up with the approximate value of upto 5 decimal places.

**What is Pythagoras theorem? **

In a simple sense, the Pythagorean theorem states that “the sum of squares of the lengths of other two sides of the** right-angled triangle** is equal to the square of the length of the hypotenuse (or the longest side)”.

In the mathematical sense, we can write Pythagoras theorem as:

Where is the side of the triangle, is the base of the triangle and is the hypotenuse.

When the condition is satisfied by the integers, then they are called **Pythagorean Triplets**.

**Proof of Pythagoras theorem – Verify,**

Application and uses of Pythagoras theorem – in real life

- Let’s take an example to explain in a better way why this theorem is important in real life

In mathematics, Pythagoras theorem is used to know if the triangle is a right-angled triangle or not. When we know the length of two sides of a triangle and the third is side is to be found out, then this theorem is used. - Suppose, you have come to the furniture showroom to buy an entertainment set but you are not sure if the TV will fit in the set or not. You just know the screen size, you can easily know the diagonal side using Pythagoras theorem and know if the TV will fit that furniture set or not.

Today, Pythagoras theorem have seen its usage in advanced math, computing surface areas, volumes, perimeters various geometric shapes etc. - Architects/engineers use this to see the correct vertical of the walls in construction.
- Oceanographers use this to determine the speed of sound in water.
- Meteorologist and aerospace find the range and sound source using the Pythagoras theorem.
- The general sorting method is found out using the theorem and is used in computer science and social networks. According to Metcalf, a network of 50M is in better coherence than a network of 70M.

(50M = 30M + 40M)

And much more.

**Let’s solve an Example using Pythagoras Theorem:**

** ****Question. Find the hypotenuse of a triangle whose lengths of two sides are:
5 cm and 12 cm.
Solution.**Using the Pythagoras theorem,

** ** Hence the hypotenuse of the triangle is 13 cm.

**List of Pythagorean Triplets:**

**Here is the list of the most commonly used Pythagorean triplets:**

**Pythagoras Theorem:**

Pythagoras Theorem can be defined as ‘In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides of the triangle’

Refer to the following figure. \(∆ABC\) is a right angled triangle, right angled at \(B\). The side opposite to the right angle is called the “hypotenuse” of that triangle. In our example, \(AC\) is the hypotenuse. \(∠A\) and \(∠C\) are the acute angles. We consider the “perpendicular side” (or “opposite side”) and the “base” (or “adjacent side”) as sides \(AB\) or \(BC\) depending on the (acute) angle we consider.

If \(∠A\) is acute angle in consideration, then \(BC\) is the “perpendicular side” (or “opposite side”) and \(AB\) is the “base” (or “adjacent side”).

If \(∠C\) is acute angle in consideration, then \(AB\) is the “perpendicular side” (or “opposite side”) and \(BC\) is the “base” (or “adjacent side”).

The statement of Pythagoras Theorem can be mathematically expressed as:

\((hypotenuse)^2\) = \((perpendicular)^2 + (base)^2\)

\(⇒AC^2\) = \(AB^2 + BC^2\)

Now, we shall try to prove this theorem using similarity of triangles. Before that, let us briefly look into the conditions for the similarity of triangles.

If two triangles are similar to each other,

- Corresponding angles of both the triangles are equal and
- Corresponding sides of both the triangles are in proportion.

Thus two triangles \(ΔABC\) and \(ΔPQR\) are similar if,

- \(∠A\) = \(∠P\), \(∠B\) = \(∠Q\) and \(∠C\) = \(∠R\)
- \(\frac{AB}{PQ}\) = \(\frac{BC}{QR}\) = \(\frac{AC}{PR}\)

Let us now consider a right angle triangle \(∆ABC\) which is right angled at \(B\).

Let \(BD\) be perpendicular to the hypotenuse \(AC\)

We can see that in \(∆ABC\) and \(∆ADB\)

\(∠ADB\) = \(∠ABC\) = \(90°\)

\(∠BCA\) = \(∠DBA\)

\(∠A\) = \(∠A\)

According to \(AA\) criterion for similarity of triangles \(∆ABC\) ~ \(∆ADB\)

Also, in \(∆BDC\) and \(∆ABC\)

\(∠CBD\) = \(∠DAB\)

\(∠BCD\) = \(∠DBA\)

\(∠ADB\) = \(∠CDB\) = \(90°\)

According to \(AA\) criterion for similarity of triangles \(∆BDC\) ~ \(∆ABC\)

Therefore, we can say that \(∆ADB\) ~ \(∆BDC\).

Thus, we can say that if we draw a perpendicular from the right-angled vertex of a right triangle to the hypotenuse, then the triangles formed on both sides of the perpendicular are similar to each other and to the whole triangle. This statement can actually be proved as a theorem which we shall be using further to prove the Pythagoras theorem.

Now, we have to prove \(AC^2\) = \(AB^2 + BC^2\). For this we drop a perpendicular \(BD\) onto the side \(AC\).

We know, \(∆ADB\) ~ \(∆ABC\)

Therefore, \(\frac{AD}{AB}\) = \(\frac{AB}{AC}\) (Condition for similarity)

Or we can say that, \(AB^2\) = \(AD~×~AC\) ……..(1)

Also as already discussed \(∆BDC\) ~ \(∆ABC\)

Therefore, \(\frac{CD}{BC}\) = \(\frac{BC}{AC}\) (Condition for similarity)

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, \(AC^2\) = \(AB^2 + BC^2\)

This is the proof of Pythagoras theorem.

This theorem can also be stated as:

The diagonal of a rectangle produces by itself the same area as produced by both of its sides (i.e. length and breadth).

The converse of the Pythagoras theorem is also true which states that, if the square of one side of a triangle is equal to the sum of squares of the other two sides then the angle which is opposite to the first side is a right angle. We will prove this in the upcoming discussions.

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