Pythagoras Theorem: Statement, Formula, Proof and applications.

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 triangle have been named Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle of 90°. The sides of a right triangle (say a, b and c) with positive integer values, when squared, are put into an equation called a Pythagorean theorem.

 Pythagoras Theorem Formula

Consider the triangle given above:

Where “a” is the perpendicular,

“b” is the base,

“c” is the hypotenuse.

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

Hypotenuse2 = Perpendicular2 + Base2 

c2 = a2 + b2  

 Pythagoras Theorem Proof

Given: A right-angled triangle ABC, right-angled at B.

To Prove- AC2 = AB2 + BC2

Construction: Draw a perpendicular BD meeting AC at D.

 

Proof:                                                                                                

We know, △ADB ~ △ABC

Therefore, 
                (corresponding sides of similar triangles) 

Or, AB2 = AD × AC                                 .……..(1)

Also, △BDC ~△ABC

Therefore,  

                (Corresponding sides of similar triangles)

Or, BC2= CD × AC                                 ………..(2)

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.

Note: Pythagorean theorem is only applicable to Right-Angled triangle.

Applications of Pythagoras Theorem

  • To know if the triangle is a Right-Angle Triangle or not.
  • Pythagoras The theorem is commonly used to find the lengths of sides of a right-angled triangle.
  • The Theorem is used to find the length of diagonal of a rectangle, square, etc.
  • Pythagoras Theorem is used in trigonometry to find the trigonometric ratios like sin, cos, tan, cosec, sec, cot.
  • Pythagoras The theorem is used in security cameras for face recognition.
  • The technique of the Pythagoras Theorem is used by architects for engineering and construction fields.
  • The Pythagoras Theorem is applied in surveying the mountains.
  • It is also used in navigation to find the shortest route.
  • By using the Pythagoras Theorem, we can derive formulas for base, perpendicular and hypotenuse.
  • Painters use ladders to paint on high buildings with the help of Pythagoras Theorem.
  • The theorem is used to calculate the steepness of slopes of hills or mountains.
  • Then the converse of Pythagoras Theorem is used to check whether a triangle is a right triangle or not

 



Comments

Post a Comment