Pythagoras Theorem – definition, formula and examples

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Pythagoras Theorem

Pythagoras theorem is a very common and important topic in mathematics. It explains the relationship between the different sides of a right triangle. The 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 of the triangle.”

The other two sides here are the perpendicular and base of the triangle. Therefore, in a right triangle, the hypotenuse becomes its longest side, because they are opposite the 90° angle.

PYTHAGOREAN THEOREM का FORMULA

In a right triangle,

A is perpendicular

B is the base

C is the hypotenuse

Therefore, according to the definition of the Pythagorean Theorem, the formula would be:

Hypotenuse²= Perpendicular² + Base²

In other words, it would be:



C²= A²+ B²

Pythagorean theorem PROOF :

In a right triangle, the base and perpendicular make an angle of 90 degrees with each other. Therefore, according to the Pythagorean theorem, “The square of the hypotenuse is equal to the sum of the square of the base and the square of the perpendicular.”
“the square of the hypotenuse is equal to the sum of a base square and perpendicular square.”

To prove this theorem,

Let us assume a triangle ABC, at right angle to angle B.

We have to prove: AC²= AB² + BC²



To explain: We draw a straight line BD which meets AC at D.



Proof:

We know by the theorem that if a right angle is drawn from the hypotenuse of a triangle at right angle, then the two triangles on both sides of the perpendicular are similar to each other.

so,

△ADB ~ △ABC

Hence,

AD/AB = AB/AC (Condition for similarity)

Or, AB2 = AD × AC (1)

Also, △BDC ~△ABC (By applying the same theorem)

Therefore,

CD/BC = BC/AC (Condition for similarity)

Or,

BC2= CD × AC (2)

Now,

By adding the equations (1) and (2) we get,

AB2 + BC2 = AD × AC + CD × AC

AB2 + BC2 = AC (AD + CD)

Since, AD + CD = AC



Therefore, AC2 = AB2 + BC2



Hence, the Pythagorean theorem is proved.

APPLICATIONS OF PYTHAGOREAN THEOREM

Some Applications of Pythagoras Theorem

  • How to know if a triangle is a right triangle or not
  • to find the diagonal of a square
  • In a right triangle, we can calculate the length of any side if we know the lengths of the other two sides.

Number Series Questions for Bank Exams

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