## Proof Pythagorean Theorem

Pythagorean theorem is a well-known geometric theorem where the sum of the squares of two sides of a right angle is equal to the square of the hypotenuse. The Pythagorean theorem has a long association with a Greek mathematician-philosopher Pythagoras and it is quite older than you may think of.

With a deep understanding of the theorem, you can solve complex geometrical problems with satisfactory answers. The whole credit for the discovery of theorem goes to Pythagoras in India. This is really surprising that concept was proposed almost 4000 years back and there are more than 400 proofs for the same that was given by the Greek mathematician Pythagoras.

There are a few people who argue that the concept was invented during 500 B.C. before the proposal of Greek mathematician Pythagoras. They are also helping to solve the hypotenuse of an isosceles triangle to compute the approximate up to five decimals.

Proof: First, we have to drop a perpendicular BD onto the side AC

We know, △ADB ~ △ABC

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

Or, \( AB^2 = AD \times AC \) ……………………………..……..(1)

Also, △BDC ~△ABC

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

Or, \( BC^2= CD \times AC \) ……………………………………..(2)

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

\( AB^2 + BC^2 = AD \times AC + CD \times AC \)

\( AB^2 + BC^2 = AC (AD + CD) \)

Since, AD + CD = AC

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

Hence, the Pythagorean theorem is proved.

As shown in the image above, the Pythagoras theorem states that the sum of the squares of two sides of a right angle is equal to the square of the hypotenuse. Where c would always be the hypotenuse.

## the proof converse of the Pythagorean Theorem

If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

That is, in ΔABC, if \( c^2=a^2+b^2 \) then ∠C is a right triangle, ΔPQR being the right angle.

**We can prove this by contradiction.**

Let us assume that \( c^2=a^2+b^2 \) in ΔABC and the triangle is not a right triangle.

Now consider another triangle ΔPQR. We construct ΔPQR so that PR=a, QR=b and ∠R is a right angle.

By the Pythagorean Theorem, \( (PQ)^2=a^2+b^2 \).

But we know that \( a^2+b^2=c^2 \) and \( a^2+b^2=c^2 \) and c=AB.

So, \( (PQ)^2=a^2+b^2=(AB)^2 \) .

That is, \((PQ)^2=(AB)^2 \).

Since PQ and AB are lengths of sides, we can take positive square roots.

PQ=AB

That is, all the three sides of ΔPQR are congruent to the three sides of ΔABC. So, the two triangles are congruent by the Side-Side-Side Congruence Property.

Since ΔABC is congruent to ΔPQR and ΔPQR is a right triangle, ΔABC must also be a right triangle.

**This is a contradiction. Therefore, our assumption must be wrong.**

### Pythagoras Formula

This is a fundamental theorem in mathematics that explains the relationship between three sides of a right-angled triangle. Till the time, we have studied the basic concept of the Pythagorean theorem. Here we will check on Pythagoras formula and its applications. A right-angled triangle is a triangle whose one side is angled at 90-degree and its opposite side is known as the hypotenuse. The other two adjacent sides of the triangle are perpendicular and the base.

According to the Pythagoras theorem, the square of the length of the hypotenuse is always equal to the sum of the square of perpendicular and the base. In the same way, you can calculate the perpendicular or base value by adjusting the values at another side. The Pythagoras formula in mathematics is written as –

\( Hypotenuse^{2}= Perpendicular^{2} + Base^{2} \)

It is used for plenty of applications in real-life like it is used to check either the given triangle is a right-angled or not. It is used in aerospace science and it is used by the meteorologists to compute the range or sound source with the help of Pythagoras theorem. Further, it can be optimized by oceanographers to check the speed of sound inside water.

### Pythagoras rule

As we know that square of the hypotenuse is equal to the sum of squares of two other sides in Pythagoras theorem then Pythagoras rule could be given as –

\( Hypotenuse^{2}= Perpendicular^{2} + Base^{2} \)

\( Base^{2}= Perpendicular^{2} – Hypotenuse^{2} \)

Where c is always the hypotenuse and don’t forget to compute the square root in the last step. Till the time we have discussed hypotenuse only i.e. c. The other two sides are perpendicular (a) and the base (b) and Pythagoras rules are applicable to the right-angled triangle only.

**Question 1**: Find the hypotenuse of a triangle whose lengths of two sides are 4 cm and 10cm.

Using the Pythagoras theorem,\( Hypotenuse^{2}= Perpendicular^{2} + Base^{2} \)

\( Hypotenuse^{2}= 10^{2} + 4^{2} \)

\( Hypotenuse^{2}= 100 + 16 \)

\( Hypotenuse^{2}= 116 \)

\( Hypotenuse = \sqrt{116} \)

\( Hypotenuse = 10.77 cm \)

**Problem 2:**The sides of a triangle are 5,12 & 13 units. Check if it has a right angle or not.

Solution: From Pythagoras Theorem, we have;

\( Hypotenuse^{2}= Perpendicular^{2} + Base^{2} \)

Perpendicular = 12 units

Base = 5 units

Hypotenuse = 13 units

\( 13^{2}= 12^{2} + 5^{2} \)

⇒ 144 + 25 = 169

⇒ 169 = 169

L.H.S. = R.H.S.