Pythagoras Theorem: Formula, Theorem, Proof and Examples

Pythagoras theorem is a geometric theorem in which the sum of squares on the foot of the right triangle is equal to the square of its hypotenuse. This theorem is used to find the length of the sides and angles of a right-angled triangle that is not known. A right-angled triangle is a polygon having 3 sides, having one of the angles as right angle (90°). Using this theorem, we can extract the perpendicular, base, and hypotenuse formulas.

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Pythagoras Theorem

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The theorem states that “The square of the hypotenuse in a right-angled triangle is equal to the sum of squares of the other two sides”. For a right-angled triangle, the Hypotenuse is considered the longest side which is opposite to the 90° angle. The other two sides referred to in the definition are called Base or adjacent and Perpendicular or the height.

The sides of a right-angled triangle (a, b, c) come with a positive integer value when they are squared and put into the equation known as the Pythagorean Triple.

The Pythagoras theorem is only applicable when the triangle is a Right-angled Triangle.

Pythagoras Theorem

The video below explains this:

Pythagoras Theorem Detailed Video Explanation:

Pythagoras Theorem Formula 

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Consider the above right-angled triangle PQR. In this triangle –

PQ= “a” is the perpendicular,

QR= “b” is the base and 

PR= “c” is the hypotenuse.

Angle Q is 90° angle

So, according to the definition given by Pythagoras, the Pythagorean Theorem Formula is given by-

Hypotenuse² = Perpendicular² + Base²

i.e. c²=a²+b² 

Consider 3 squares a, b, c on three sides of a triangle as shown in the figure below.

Now, by Pythagoras Theorem-

Pythagoras Theorem Formula

Area of square “c” = Area of square “a” + Area of square “b”.

Note: “a” and “b” are the other two sides and “c” is the longest axis.

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Proof of Pythagoras Theorem

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Let us consider a Right-angled triangle PQR, right-angled at Q.

To Prove: PR²=PQ²+QR²

Construction: Draw a perpendicular QS meeting at PR at S.

A perpendicular QS meeting at PR at S

Proof: 

We know, ΔRSQ~ΔRQP

Therefore, RSRQ=RQRP (Corresponding sides of Similar Triangle)

Or, RQ²=RS * RP…………………..(1)

Also, ΔQSP~ΔRQP

Therefore, PSQP=QPRP (Corresponding sides of Similar Triangle)

Or, QP²=PS * RP…………………..(2)

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

RQ² + QP²= RS * RP + PS * RP 

RQ² + QP²= RP( RS+PS)

Since RS+PS=RP

Therefore, RP²=RQ²+QP² 

Or, c²= b²+a²

Or, c² =a²+b²

Or, Hypotenuse² = Perpendicular² + Base²

Hence, The Pythagoras Theorem is proved.

The Converse of Pythagoras Theorem

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The converse of Pythagoras theorem can be stated as: A triangle can be proved to be a right-angle triangle when 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. 

In \(\bigtriangleup\) ABC, if C² = a² + b², then \(\angle\)C is a right angle triangle, \(\bigtriangleup\)PQR being the right angle

The Converse of Pythagoras Theorem

This can be proved by contraction

Let’s assume c² = a^{2 +} b² in \(\bigtriangleup\)ABC and the triangle is not right triangle.

Then let's assume another \(\bigtriangleup\)PQR. Through the construction of \(\bigtriangleup\)PQR so that PR = a, QR = b and \(\angle\)R is a right angle.

The Converse of Pythagoras Theorem

By the Pythagoras theorem, (PQ)² = a² + b²

But we are aware that a² + b² = c² and c = AB

Hence (PQ)² = a² + b² = (AB)²

That is (PQ)² = (AB)²

Since PQ and AB are the lengths of the sides, we can consider them to be positive square roots

PQ = AB

That is all the three given sides of \(\bigtriangleup\)PQR is congruent to all the three sides of \(\bigtriangleup\)ABC. Hence the two given triangles are congruent by the Side-Side-Side property of congruency.

Since the \(\bigtriangleup\)ABC is congruent to \(\bigtriangleup\)PQR and it is a right angle triangle, therefore \(\bigtriangleup\)ABC is also a right-angle triangle.

This is a contradiction, hence the assumptions we made must be wrong. 

Application of Pythagoras Theorem

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• This theorem helps us to find the diagonal of a square.
• Pythagoras Theorem helps us to find whether a triangle is a right-angled triangle or not.
• For a Right-angled triangle using this Theorem, we can find the length of the unknown side of the other two sides are known
• Pythagoras theorem enables us to calculate the diagonal length of a roof, the distance between the foot of a slanted ladder, the perpendicular height of the wall or bean on which the ladder is slanted.
• Pythagoras theorem is very useful for civil engineering projects
• This theorem is also used for navigation purposes as it is difficult to measure the distance across water bodies.
• Cartographers also use the Pythagoras theorem to find the distance between two points on a 2D sheet or map
• Pythagoras theorem can also be used to find the steepness of the hill slope or mountain slope

Things to Remember

• Pythagoras theorem:  \(c = \sqrt{a^2+b^2}\)
• Pythagoras theorem states the relation between all the three sides of a right triangle.
• It helps in the calculation of the sides and angles of a triangle.
• Pythagoras theorem helps in understanding whether a triangle is a right-angled triangle or not.
• It helps in the trigonometry ratio calculation along with distance and height measurement.

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