To further extend my investigation, I am going to investigate the family of Pythagorean Triples where all three sides are positive integers and the shortest side is an even number. I will start with the shortest side being 6cm and progress until I have 10 readings.
‘B+2’
I am not going to draw any triangles from now on because I find it much easier and quicker to use trial and improvement to find the results for my table than it would drawing 10 triangles. Here is my results table for ‘B+2’.
From this table, I have found a rule to find Side A, a rule to find Side B and rule to find Side C.
Rules For B+2
Side A - 2n+4. This is explained by two multiplied by the term number plus four. For example, if the term number is three, then Side A would be ten.
Side B - n²+4n+3. This is explained by squaring the term number, adding 4 multiplied by the term number and adding 3.
Side C - B+2. This is explained by Side B plus 2. For example, if Side B is 5, Side C would be 7.
To extend my investigation, I am going to try and find overall rules for sides A, B, and C. I will have separate rules for when side a is an odd number and when side A is an even number. To find overall rules, I will need to continue finding Pythagorean Triples for when Side C is ‘B+3’ and then ‘B+4’ until I am up to ‘B+10’. When I reach this point, I will have sufficient information to find my overall rules.
‘B+3’
Here is my results table for ‘B+3’.
From this table, I have found a rule to find Side A, a rule to find Side B and rule to find Side C.
Rules For B+3
Side A - 6n+3. This is explained by six multiplied by the term number plus three. For example, if the term number is four, then Side A would be twenty-seven.
Side B - n(6n+6). This is explained by 6 multiplied by the term number then plus six, this answer is then multiplied by the term number.
Side C - B+3. This is explained by Side B plus 3. For example, if Side B is 5, Side C would be 8.
‘B+4’
Here is my results table for ‘B+4’.
From this table, I have found a rule to find Side A, a rule to find Side B and rule to find Side C.
Rules For B+4
Side A - 4n+8. This is explained by four multiplied by the term number then plus eight. For example, if the term number is four, then Side A would be twenty-four.
Side B - 2(n²+4n+3). This is explained by squaring the term number, adding 4 multiplied by the term number, adding 3 and then multiplying that number by 2.
Side C - B+4. This is explained by Side B plus 4. For example, if Side B is 4, Side C would be 8.
‘B+5’
Here is my results table for ‘B+5’.
From this table, I have found a rule to find Side A, a rule to find Side B and rule to find Side C.
Rules For B+5
Side A - 10n+5. This is explained by ten multiplied by the term number then plus five. For example, if the term number is one, then Side A would be fifteen.
Side B - an+5n. This is explained by Side A multiplied by the term number then plus five times the term number. For example, if the term number was 2 and side A was 25, Side B would be 60.
Side C - B+5. This is explained by Side B plus 5. For example, if Side B is 10, Side C would be 15.
‘B+6’
Here is my results table for ‘B+6’.
From this table, I have found a rule to find Side A, a rule to find Side B and rule to find Side C.
Rules For B+6
Side A - 6n+6. This is explained by six multiplied by the term number then plus six. For example, if the term number is one, then Side A would be fifteen.
Side B - 3(n²+4n+3) This is explained by squaring the term number, adding 4 multiplied by the term number, adding 3 and then multiplying that number by 3.
Side C - B+6. This is explained by Side B plus 6. For example, if Side B is 10, Side C would be 16.
‘B+7’
Here is my results table for ‘B+7’.
From this table, I have found a rule to find Side A, a rule to find Side B and rule to find Side C.
Rules For B+7
Side A - 14n+7. This is explained by fourteen multiplied by the term number then plus seven. For example, if the term number is one, then Side A would be twenty-one.
Side B - an+7n. This is explained by Side A multiplied by the term number then plus seven times the term number. For example, if the term number was 2 and side A was 35, Side B would be 84.
Side C - B+7. This is explained by Side B plus 7. For example, if Side B is 10, Side C would be 17.
‘B+8’
Here is my results table for ‘B+8’.
From this table, I have found a rule to find Side A, a rule to find Side B and rule to find Side C.
Rules For B+8
Side A - 8n+16. This is explained by eight multiplied by the term number then plus eight. For example, if the term number is one, then Side A would be twenty-four.
Side B - 4(n²+4n+3) This is explained by squaring the term number, adding 4 multiplied by the term number, adding 3 and then multiplying that number by 4.
Side C - B+8. This is explained by Side B plus 8. For example, if Side B is 10, Side C would be 18.
‘B+9’
Here is my results table for ‘B+9’.
From this table, I have found a rule to find Side A, a rule to find Side B and rule to find Side C.
Rules For B+9
Side A - 18n+9. This is explained by eighteen multiplied by the term number then plus nine. For example, if the term number is one, then Side A would be twenty-seven.
Side B - an+9n. This is explained by Side A multiplied by the term number then plus nine times the term number.
Side C - B+9. This is explained by Side B plus 8. For example, if Side B is 36, Side C would be 45.
‘B+10’
Here is my results table for ‘B+10’.
From this table, I have found a rule to find Side A, a rule to find Side B and rule to find Side C.
Rules For B+10
Side A - 10n+20. This is explained by ten multiplied by the term number then plus twenty. For example, if the term number is one, then Side A would be thirty.
Side B - 5(n²+4n+3). This is explained by squaring the term number, adding 4 multiplied by the term number, adding 3 and then multiplying that number by 5.
Side C – B+10. This is explained by Side B plus 10. For example, if Side B is 10, Side C would be 20.
Formulas
I will now add all of my formula’s into a table to see if there are any overall rules.
I have noticed that in the above table there are two different patterns. The two patterns are one when ‘B+…’ is even and when ‘B+…’ is an odd number. Because of this, I will now draw two separate tables.
Odds
Evens
Overall Rules
Key – X = B+… (whatever side c is)
Odds
Side A – 2X + X. This is explained by 2 times by Side C plus Side C.
Side B – AN + XN. This is explained by Side A times the term number, plus Side C times by the term number.
Area – ½ A X B. This is explained by 0.5 times Side A times Side B.
Perimeter – A+B+C. This is explained by Side A plus Side B plus Side C.
Evens
Side A – XN + 2X. This is explained by Side C times the term number, plus two time Side C.
Side B – (0.5 X)(n²+4n+3). This is explained by 0.5 times Side C, times by the term number squared plus four times the term number plus 3.
Area – ½ A X B. This is explained by 0.5 times Side A times Side B.
Perimeter - A+B+C. This is explained by Side A plus Side B plus Side C.