BLOG ENTRY #4

Im investigating the question WHILE MEASURING A TRIANGLE WHEN CAN YOU TELL THAT THE LENGTHS ARE A SET OF PYTHAGOREAN TRIPLES? IF SO IS HIS A MATH THING THAT YOU ARE JUST GOING TO HAVE TO REMEMBER ALL OF THEM OR IS THERE A WAY YOU CAN FIGURE IT OUT USING OTHER METHODS? like for example can you use the diaphantus formula to prove that its one of many Pythagorean Triples?
I STARTED INVESTIGATING BY LOOKING AT A TRIANGLE THAT HAD THE SIDE LENGTHS 3-4-5 WHICH IS AN PYTHAGOREAN TRIPLE. TO GO ABOUT INVESTIGATING MY QUESTION ALONG WITH THIS TRIANGLE I WENT BACK TO THE DIOPHANTUS FORMULA. WHEN I USED THE FORMULA TO FIND THIS TRIPLE MY CALCULATIONS WERE:
1*2*(2)= 4
1^2+1^2= 1+4= 5
2^2-1^2= 4-1= 3
IF I DIDNT KNOW THAT 3-4-5 WAS A TRIPLE AND ALL I WAS GIVEN WAS TWO LENGTHS HOW WOULD I KNO THAT THIS WOULD B A TRIPLE?
WELL THATS WHAT I WANT TO KNO ND BEEN INVESTIGATING AND HAVEN'T REALLY GOTTEN ANY WHERE. IF I WERE GIVEN SIDES "A" AND "B" WHICH IS 3 AND 4 HOW CAN I FIND OUT THE HYPOTENUSE? WELL THAT WAS A QUESTION THAT WAS EASY FOR ME TO ANSWER BUT THATS WHERE I BEGAN. THE FORMULA THAT WILL HELP ME WITH THAT ANSWER WAS THE PYTHAGOREAN THEREOM WHICH IS:
a^2+b^2= c^2
3^2+4^2= 5^2
SEE ALREADY KNEW THIS SO I THOUGHT SOME MORE. I CAME UP WITH THIS IDEA AS TO IF IT IS POSSIBLE TO WORK THE DIOPHANTUS FORMULA BACKWARDS. WELL I DIDNT KNO HOW TO GO ABOUT DOING THAT SO I REARRANGED THE PROCESS. I KEPT THE 1ST STEP:

• 1*2*(2)= 4

THEN INSTEAD I DID STEP 3 SECOND:

• 2^2-1^2= 4-1= 3

THIS WAY I HAD SIDE LENGTHS "a" AND "b"
I THOUGHT ABOUT IT AND SAID I CAN GET THE HYPOTENUSE 2 DIFFERENT WAYS. I COULD HAVE DID STEP 2 OF THE DIOPHANTUS FORMULA:

• 1^2+1^2= 1+4= 5

OR I COULD HAVE USED THE FORMULA FOR THE PYTHAGOREAN TRIPLE:

• a^2+b^2= c^2

• 3^2+4^2= 5^2

I NOTICED THAT EITHER FORMULA WOULD HAVE HELPED ME GET THE RESULTS THAT I WANTED.
BUT THIS DID LEAVE ME WITH ANOTHER QUESTION IN MY MIND. HOW IS THE DIOPHANTS FORMULA RELATED TO THE PYTHAGOREAN THEROEM FORMULA IF THERE ARE ANY RELATIONS AMOUNG THE TWO AT ALL?