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Pythagoras Theorem: Formula, Proof, Examples And Applications

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.

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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.

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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”

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

AB² + BC² = AC²

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Pythagoras 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)

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∠ 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 Applications

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The applications of Pythagoras 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.

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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

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Also, BC = 2.5 m and CA = 6 m

From Pythagoras 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.

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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.

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