Pythagoras Theorem: Pythagoras Theorem Statement and Formula (For CBSE, ICSE, IAS, NET, NRA 2022)

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Pythagoras Theorem is an important topic in Maths, which explains the relation between the sides of a right-angled triangle. It is also sometimes called Pythagorean Theorem. The formula and proof of this theorem are explained here. This theorem is basically used for the right-angled triangle.

Pythagoras Theorem Statement

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides” . The sides of this triangles have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say x, y and z) which has positive integer values, when squared are put into an equation, also called a Pythagorean triple.

Pythagorean Theorem Statement

Example:

The examples of theorem based on the statement given for right triangles is given below:

Consider a right triangle, given below:

Pythagorean Theorem Statement

Find the value of x.

X is the side opposite to right angle; hence it is a hypotenuse.

Now, by the theorem we know;

Therefore, we found the value of hypotenuse here.

Pythagoras Theorem Formula

Consider the triangle given above:

Where is the perpendicular side

is the base,

is the hypotenuse side.

According to the definition, the Pythagoras Theorem formula is given as:

The side opposite to the right angle (90°) is the longest side (known as Hypotenuse) because the side opposite to the greatest angle is the longest.

Pythagorean Theorem Formula

Consider three squares of sides mounted on the three sides of a triangle having the same sides as shown.

By Pythagoras Theorem –

Pythagoras Theorem Proof

Given: A right-angled triangle ABC.

To Prove-

Pythagorean Theorem Proof

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

We know,

Therefore, (Condition for similarity)

Or,

Also,

Therefore, (Condition for similarity)

Or,

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

Since,

Therefore,

Hence, the Pythagorean theorem is proved.

Applications of Pythagoras Theorem

  • To know if the triangle is a right-angled triangle or not.
  • In a right-angled triangle, we can calculate the length of any side if the other two sides are given.
  • To find the diagonal of a square.

Pythagorean Theorem Problems

Problem 1: The sides of a triangle are . Check if it has a right angle or not.

Pythagorean Theorem Problems

Solution: From Pythagoras Theorem, we have,

Perpendicular

Base

Hypotenuse

Therefore, the angles opposite to the 13-unit side will be at a right angle.

Problem 2: The two sides of a right-angled as shown in the figure. Find the third side.

Pythagorean Theorem Problems

Solution: Given,

Perpendicular

Base

Hypotenuse

As per the Pythagorean Theorem, we have;

Therefore,

Problem 3: Given the side of a square to be Find the length of the diagonal.

Solution- Given;

Sides of a square

Pythagorean Theorem Problems

To Find- The length of diagonal ac.

Consider triangle (or can also be )

or

Thus, the length of the diagonal is

Frequently Asked Questions on Pythagoras Theorem

What is the Formula for Pythagorean Theorem?

The formula for Pythagoras, for a right-angled triangle, is given by;

What is the Formula for Hypotenuse?

The hypotenuse is the longest side of the right-angled triangle, opposite to right angle, which is adjacent to base and perpendicular. Let base, perpendicular and hypotenuse are a, b and c respectively. Then the hypotenuse formula, from the Pythagoras statement, will be;

Can We Apply the Pythagoras Theorem for Any Triangle?

No, this theorem is applicable only for the right-angled triangle.