Using Pythagoras Theorem Prove

In any right triangle, the area of the square with side hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle.

                                          

                                          Right angle triangle 

   In equation form, the theorem can be written as

         a2 + b2 = c2

   Where, a is the length of opposite side
             b is the length of adjacent side
             c is the length of the hypotenuse

 

Properties of a Pythagoras Theorem:

 

  • The area of the Pythagoras theorem is an even congruent number
  • In every Pythagorean triple, the radii of the three excircles and the radius of the incircle are natural numbers
  • The Pythagorean triple , the hypotenuse side and the other two sides are differed exactly by 1.  
  • The hypotenuse of Pythagorean triangle exceeds the even leg by the square of odd integer
  • It exceeds the odd leg by the square of an integer

 

 

Sample Problems by using Pythagoras theorem:

 

Pro 1:

Sol : Using Pythagoras theorem prove that the following triangle is a right-angled triangle.

 

                           Right-angled triangle.                          

Proof:   By using Pythagoras theorem we know that

                   Hypotenuse2 = opposite2 + adjacent2

                                    c2 = a2 + b2

  Given a = √5, b = √3 and c = √8

  Therefore by the theorem,

   (√8)2 = (√5)2 + (√3)2

  8 = 5 + 3

  8 = 8

Hence proved.

 

Problem 2:

 

Prob : Given a right triangle whose sides are: hypotenuse = 5; opposite = 3; adjacent = 4.

Using Pythagoras theorem find what type of triangle?.

Sol :   By the theorem,  

                   Hypotenuse2 = opposite2 + adjacent2

                                   52 = 32 + 42

                               25 = 9 + 16

                                25 = 25

Hence proved.

Pro3:  Using Pythagoras theorem prove the following equation.

sec θ = tan θ / sin θ

Proof:  Given sec θ = tan θ / sin θ

Squaring on both sides , we get

 sec2 θ = tan2 θ / sin2 θ

           = (sin2 θ / cos2 θ) / sin2 θ

           = 1 / cos2 θ

sec2 θ = sec2 θ

Hence proved. 

Prob 4:  Find the value of the missing length of the leg of a right triangle SAM. 

                       

                                                             Right-angled triangle.                          

Sol :   By using Pythagoras theorem

               Hypotenuse2 = opposite2 + adjacent2  

    c= 162 + 122

        = 256 + 144

    c2 = 400

      c = √400 = 20