Pythagoras Theorem, Formula, Proof, Examples And Applications

Pythagoras Theorem, Formula, Proof, Examples And Applications

Quantitative Aptitude is the important scoring section in many competitive exams and Pythagoras theorem is one of the important topics in it. Here we are providing you winformation Pythagoras Theorem.

Pythagoras Theorem

Pythagoras theorem: The Pythagoras theorem is an important concept used while solving questions in Mathematics. The theorem explains the relation between the sides of a triangle. The Pythagoras theorem was derived by Pythagoras, a Greek philosopher of the sixth century B.C who declared it as an essential property of the right-angled triangles. The property is, hence, named after him. If any triangle holds the Pythagoras property, then it is certainly a right-angled triangle. Let’s take a look at the statement, formulas, proof, applications and examples of the Pythagoras theorem.

Pythagoras Theorem: Statement & Formula

In a right-angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are known as the legs of the right-angled triangle. The hypotenuse is the longest side and the other 2 sides are named Perpendicular and base. 

Statement of Pythagoras theorem: The Pythagoras theorem states that “In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides”

From the figure given above, the Pythagoras theorem formula can be derived as:

AB² + BC² = AC²

Pythagoras’s Theorem with Proof

We are given a right triangle ABC right angled at B. Let BD be perpendicular to the hypotenuse AC. You must know the theorem “If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other”

Let us draw BD ⊥ AC (See Fig)

∠ A = ∠ A

and ∠ ADB = ∠ ABC

So, ∆ ADB ~ ∆ ABC

Similarly, ∆ BDC ~ ∆ ABC

Now, ∆ ADB ~ ∆ ABC

So, AD/AB = AB/AC (Sides are proportional)

or, AD. AC = AB²  (1)

Also, ∆ BDC ~ ∆ ABC

So, CD/BC = BC/AC or CD . AC = BC²   (2)

Adding (1) and (2),

AD. AC + CD . AC = AB² + BC²

or, AC (AD + CD) = AB² + BC²

or, AC. AC = AB² + BC²

or, AC² = AB² + BC²

Pythagoras Theorem Application

The applications of Pythagoras’s theorem can be as follows:

To solve questions based on triangles and to find if the triangle is a right-angled triangle or not.

To calculate the diagonal of a square.

This theorem is used in architecture, woodworking, and other physical construction projects

Pythagoras Theorem: Triplets

You must know the Pythagoras triplets to apply this theorem to various concepts. Some of the Pythagoras triplets are given below:

3, 4, 5

5, 12, 13

7, 24, 25

8, 15, 17

9, 40, 41

11, 60, 61

12, 35, 37

20, 21, 29

 Pythagoras Theorem Examples and Questions

Example 1: ∆ ABC is right-angled at C. If AC = 5 cm and BC = 12 cm find the length of AB.

Refer to the figure given on the right.

As the triangle is right-angled, by Pythagoras’ theorem,

AB² = AC² + BC²

AB² = 5² + 12²

AB² = 25 + 144

AB² = 169 = 13²

Hence, AB = 13 cm

Example 2: A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches a window 6 m above the ground. Find the length of the ladder.

Let AB be the ladder and CA be the wall with the window at A

Also, BC = 2.5 m and CA = 6 m

From Pythagoras’s Theorem, we have:

AB² = AC² + BC²

AB² = 6² + (2.5)² = 42.25

So, AB = 6.5

Thus, the length of the ladder is 6.5 m.

Example 3: Determine whether the triangle whose lengths of sides are 3 cm, 4 cm, and 5 cm is a right-angled triangle.

Solution: 3² = 9, 4² = 16 and 5² = 25

We find, 3² + 4² = 5²

Therefore, the triangle is right-angled.

Example 4: The side of a square is given to be 4 cm. Find the length of the diagonal of a square.

Solution: We need to find the length of the diagonal AC. As we know, all sides of a square are equal and each angle is 90 degrees, ADC is a right-angled triangle.

By Pythagoras theoram,

AD² + CD² = AC²

AC² = 4² + 4² = 16 + 16

AC² = 32, So, AC= 4√2

The diagonal of the square is 4√2 cm..