Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

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 
https://dinesh51.blogspot.com

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 δΈ„ AC

Proof:- InADB and ABC
1 = 3
A = A
            By AA rule
ADB~△ABC
    AB x AB =AC x AD
                                  AB2 = AC x AD  ................(1)
           InBCD 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)
                    AB+ BC2  = AC X AD + AC X CD

                                  AB+ BC2  = AC(AD+CD)

                                 AB+ BC2   = AC X AC

                                 AB+ BC2   =  AC2
Hence prove the required theorem
*********************************************

PROOF OF CONVERSE OF PYTHAGORAS THEOREM

https://dinesh51.blogspot.com

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,  AB+ 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

                         DE+ EF2   =  DF2  

Putting AB = DE and BC = EF we get

                        AB+ BC2   =  DF2   ..........................(1)
                But  AB+ BC2   =  AC2  ..........................(2)Given

From (1) and (2) we get

                       AC2 = DF2
               Or     AC = DF

Now InABC 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
πŸ™

Comments

Popular Post on this Blog

Lesson Plan Maths Class 10 | For Mathematics Teacher

Lesson Plan Math Class X (Ch-8) | Trigonometry

Lesson Plan Maths Class X (Ch-5) | Arithmetic Progression