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Beyond Pythagoras Investigation

explanatory Essay
4849 words
4849 words
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Beyond Pythagoras Investigation

In this investigation, I am trying to find the rules and patterns for

right-angled triangles. Each length of the right-angle triangle is

going to be a positive integer, and the shortest length is going to be

an odd number.

Below is a typical type of a Pythagoras triple equation:

[IMAGE]

Formula:a² + b² = c²

3² + 4² = 5²

9 + 16 = 25

In this equation, the first step I took was to put the numbers instead

of the letters. Once I did that I squared all the numbers, and when I

add a² + b² it equaled to c². Incase one of the numbers was missing on

the triangle; I would replace that as x², and carry out the formula as

a normal algebraic expression. When you look at the numbers at first

they don't add up. If you do 3 + 4 it does not equal to 5, but once

the numbers are squared they add up.

This is the second type of a Pythagoras triple equation:

[IMAGE]

In this example the exact same procedure was done. I put all the

numbers instead of the formula and worked it out. The first step I

took was to square all the numbers. To check that my work was correct

I added a² + b² to get c². So far both example that were shown satisfy

the condition because when I added a² + b² I got c². One thing I found

in common between both of the right-angled triangle was that the

hypotenuse and the second biggest number had a one number difference.

This is the third type of Pythagoras triple:

[IMAGE]

On this third example, the Pythagoras triple worked because once 'a'

and b were squared, and when added then they equal to c². Just like

the other two examples, I used the same method to solve the equation

and also the check it. Also, in this example there are similar things

between the hypotenuse and the second biggest side, just like in the

In this essay, the author

  • Explains that 3 + 4 does not equal to 5, but once. they don't add up.
  • Opines that the next step is to take a closer look at each side of the document.
  • Explains that when they did that, they first did 2*1, which gave them 2, and then they had to summary:
  • Explains that this time they made 'n' equal to 7. their formula was: 2*7 = 14, and when
  • Explains that the third step was to find out what n2 was.
  • Narrates how the last step was to get the answer.
  • Narrates how they had 2n, which turned out to be 2n.
  • Opines that they would like to see how much more or less they need to get to.
  • Opines that now that they have found out this part, they just need to put all their formulas.
  • Describes the steps to keep the same formula.
  • Explains that in order to do that, they will carry out the same steps.
  • Explains that all they need to do is write out all three formulas in one.
  • Explains that there are two checks below to show that the result turns out to be.
  • Describes the dimensions of a perimeter and explains how they are calculated.
  • Explains that perimeter = (2*8 + 1) + (2)*82 + 2*8) +
  • Explains that if you look back at the result table you can see the answer is summary:
  • Explains that they added all the like terms together to make a complete list.
  • Explains that perimeter = 56 units. if you look at the table, you can see that this is true.
  • Explains that the formula does work even if it is not.
  • Explains that area = 330 units squared. if you look back at the result table, you can see the area.
  • Narrates how they added up the common numbers and kept one thing in mind.
  • Describes a fact that we must keep in mind: this can only work when all requisites are met.
  • Opines that there is no table to refer to, but the only way we can find out is if it exists.
  • Narrates how they multiplied 2n and 1, which gave them 4n2.
  • Opines that at the moment, there is no way to find out if it is correct.
  • Describes the 4n4+4n3 + 2n2 + 4nd +4nd+2n +2nd.
  • Explains that with all these answers in the box, they can now simplify.
  • Explains that a2 + b2 = c2 format. however, this time there will be minor changes.
  • Explains that they will try this twice, in order to have a fair result.
  • Describes the values of (4*32 + 4*3 + 1) + (4 *34 + 8*33 + 4-32).
  • Explains that when 49 was added with 576, as an answer, we got 49.
  • Opines that 2n + 1) + (2n2+1) wouldn't equal (n = 2), but since they're not equal, they would.
  • Explains that (4*72 + 4*7 + 1) + ((4)*74 + 8*73 + 4-72) = (44*75 + 8)
  • Explains that they used the same technique as above to find out the numbers from 1 to 10.
  • Explains that it was one of the main steps they had to take to simplify them.
  • Explains that the first set of odd numbers was: 3, 4, and 5, so when i multiplied.
  • Describes 3 * 2 = 6, 4 * 2, 8, and 5 *2 = 10.
  • Describes the even numbers of 5 * 2, 10, 12 * 2 = 24, and 13 *2 = 26.
  • Describes the values of 7 * 2 = 14, 24, and 25*2 = 50.
  • Explains that these were the even numbers that they found, but to find out where they came from.
  • Explains that they had to multiply 4n + 2 in order to find this out.
  • Describes the formula for the middle side and explains how they will carry out the summary.
  • Explains how they multiplied 2 by 2n, which gave them an answer of 4n.
  • Explains that the first set of numbers i got was 6, which is also true.
  • Opines that just like before, they will carry out one more check to make.
  • Describes the table i made using the third set of odd.
  • Explains that they will make a table with n ranging from 1 to 10.
  • Explains that they had to multiply 4n2+4n to find this out.
  • Explains that the largest side is by two, so the formula would be 4n2+4n + 2.
  • Explains that the formula for the odd set of numbers was 2n2+2n + 1.
  • Explains that they will do two checks to check that this is correct.
  • Explains how they got the hypotenuse when using the third set of odd.
  • Explains that they will be using the table above, but they are still going to find out a summary.
  • Explains that in order to simplify this formula, all they have to do is simplify the formula.
  • Explains that the perimeter is 8n2 + 12n + 4. the next part will be to check if this is the case.
  • Explains that they can check and see if their answers are correct.
  • Explains the first bracket, which is (4n + 2) by 0.5.
  • Explains that the first part is complete, and the second part will be completed.
  • Explains that there is an easy way to simplify this, which is (4n2 + 4n).
  • Explains that with each of the expressions they found out i have to square.
  • Describes the steps they will take to simplify the process.
  • Narrates how they did 4n by 2, which gave them 8n.
  • Explains that at the moment there is no way to check if it is correct.
  • Describes the middle side = 16n4+16n3 + 8n2 + 1631628n+82.
  • Describes 16n2+16n + 4) + (16n4+32n3 + 16
  • Describes the values of (16*42 + 16*4 + 4) + (16 *44 + 32*43 +
  • Explains that it equaled to the longest side.
  • Describes the factors that make up the difference between (16*72 + 16*7 + 4) and (16*74 + 32*73 +
  • Explains how they squared all the numbers, and when they add a2 + b2, it equaled to c2.
  • Explains that they added a2 + b2 to get c2, and both examples that were shown satisfy the condition.
  • Explains that when they use the formula above, they do not end up with the answer they should get.
  • Narrates how they worked out the formula for 'n' and checked to see if it was on the result table.
  • Explains that the shortest side formula is 2n + 1, followed by the middle-side formula which is 2,n2 + 2. for the longest side, there is no point in.
  • Explains that for the 2n2, you can times it by 2, or just add it to the 4n2. when that is done, you need to add 1 by 1.
  • Explains the formula of (2n + 1) 2 + (2n2 + 2) 2 = (n2) + 2n2.
  • Illustrates how the numbers in a bracket are arranged.
  • Explains that on the table above, the same steps were taken just like the previous table, except for one bracket.
  • Explains that the largest number is 4n4 + 8n3 +8n2 + when all the like terms are added together.
  • Explains that in order to check if this works out, they will replace n as an integer.
  • Explains that the formula is correct, and it can be used when n is.
  • Explains that their formula for the shortest side was 4n + 2.
  • Explains that the shortest side = 6 gave them the answer to one of their questions, which was where would the even set of numbers i found go.
  • Explains that by doing this check, they also found out what n would have to equal to in order for them to find out where the numbers were.
  • Explains how they add up all the 4n in the equation and multiply it by 3 to get 12n.
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