Proof of Pythagoras Theorem and its Converse

PYTHGORAS THEOREM AND ITS CONVERSE
CLASS 10 CHAPTER 6  

Proof of Pythagoras theorem and its converse for class X, complete explanation of the Pythagoras theorem and its converse, Statement and proof of Pythagoras theorem class x, statement and proof of converse of Pythagoras theorem. 

PROOF OF PYTHAGORAS THEOREM  STATEMENT:- In a right angled triangle sum of square of two sides of a triangle is equal to the square of the third side. Given:- In triangle ABC, ∠B = 90o To Prove :- AB2 + BC2 = AC2 Construction: Draw BD 丄 AC Proof:- In△ADB and △ABC ∠1 = ∠3 ∠A = ∠A             ∴ By AA ～ rule △ADB～△ABC     AB x AB =AC x AD                                   AB2 = AC x AD  ................(1)            In△BCD and △ACB ∠2 = ∠3 ∠C = ∠C           ∴ By AA ～ rule △BCD～△ACB      BC x BC =AC x CD                                  BC2 = AC x CD  ..............(2)              Adding equation (1) and (2)                     AB2 + BC2  = AC X AD + AC X CD                                   AB2 + BC2  = AC(AD+CD)                                  AB2 + BC2   = AC X AC                                  AB2 + BC2   =  AC2 Hence prove the required theorem ********************************************* PROOF OF CONVERSE OF PYTHAGORAS THEOREM STATEMENT:- If sum of squares of two sides of a triangle is equal to the square of the third side then the angle opposite to the larger side is right angle. GIVEN :- In triangle ABC,  AB2 + BC2   =  AC2 TO PROVE:-  ∠B = 90o CONSTRUCTION:-  Draw another △ DEF such that  AB = DE,  BC = EF, and ∠E = 90o PROOF:-  In △DEF , ∠E = 90o Therefore by Pythagoras Theorem                          DE2 + EF2   =  DF2   Putting AB = DE and BC = EF we get                         AB2 + BC2   =  DF2   ..........................(1)                 But  AB2 + BC2   =  AC2  ..........................(2)Given From (1) and (2) we get                        AC2 = DF2                Or     AC = DF Now In△ ABC and △DEF                      AB = DE   ..................(By construction)                      BC = EF .....................(By construction)                      AC = DF ......................(Proved ) ∴ By SSS ≌ rule In △ABC ≌ △DEF                           ∠B = ∠E  ...............(By CPCT)                   But ∠E = 90o                       ∴ ∠B = 90o Hence prove the required result THANKS FOR WATCHING AND READING 🙏