Saneeta Mandil 10R
Maths Coursework
Beyond Pythagoras
Pythagoras Theorem is a2 + b2 = c2. 'a' being the shortest side, 'b' being the middle side and 'c' being the longest side (hypotenuse) of a right angled triangle.
Pythagorean triples are a family of right-angled triangles for which all the positive integers and the shortest side is an odd number.
The first part of my investigation is to study the family of Pythagorean triples where the shortest side is an odd number and all three sides are positive integers.
The numbers 3, 4 and 5 satisfy the condition
32 + 42 = 52
(Smallest number) 2 + (middle number) 2 = (largest number) 2
The numbers 3, 4 and 5 can be the lengths - in appropriate sides units - of the sides of a right-angled triangle.
5
3
4
32 = 3 x 3 = 9
42 = 4 x 4 =16
52 = 5 x 5 =25
and so 32+ 42 = 9 + 16 =25
The numbers 5, 12 and 13 can also be the length in appropriate units of a right-angled triangle.
3
5
12
52 = 5 x 5 = 25
22 = 12 x12 =144
32 = 13 x 13 =169
And so 52 + 42 = 25 + 144 =169
This is also true for the numbers 7, 24 and 25.
25
7
24
7 = 7 x 7 = 49
24 =24 x 24 = 576
25 = 25 x 25 = 625
and so 7 + 24 = 49 + 576 =25
(3,4,5), (5,12,13) and (7,24,25) are called Pythagorean triples because they satisfy the condition.
The perimeter and area of the triangle with lengths 3, 4,5 are
Perimeter = 3 + 4 + 5 = 12 units
Area = 1/2 x 3 x 4 = 6 square units
I have put these Pythagorean triples (3,4,5), (5,12,13) and (7,24,25) into a table in which I have noticed many patterns occurring.
n
a
b
c
Perimeter
Area
3
4
5
2
6
2
5
2
3
30
30
3
7
24
25
56
84
Patterns
* 'a' increases by + 2 each term.
* 'a' is equal to term number times 2 then add 1.
* The last digit of 'b' is in the pattern of 4,2,4.
* The last digit of 'c' is in the pattern of 5,3,5.
* The square root of (b + c) = a.
* 'c' is always + 1 to 'b'.
* 'b' increases by + 4 each term
* ('a' x 'n') + n = 'b'
From the first three Pythagorean triples, I have found formulae for the sides of a, b, c, and the perimeter and area.
Maths Coursework
Beyond Pythagoras
Pythagoras Theorem is a2 + b2 = c2. 'a' being the shortest side, 'b' being the middle side and 'c' being the longest side (hypotenuse) of a right angled triangle.
Pythagorean triples are a family of right-angled triangles for which all the positive integers and the shortest side is an odd number.
The first part of my investigation is to study the family of Pythagorean triples where the shortest side is an odd number and all three sides are positive integers.
The numbers 3, 4 and 5 satisfy the condition
32 + 42 = 52
(Smallest number) 2 + (middle number) 2 = (largest number) 2
The numbers 3, 4 and 5 can be the lengths - in appropriate sides units - of the sides of a right-angled triangle.
5
3
4
32 = 3 x 3 = 9
42 = 4 x 4 =16
52 = 5 x 5 =25
and so 32+ 42 = 9 + 16 =25
The numbers 5, 12 and 13 can also be the length in appropriate units of a right-angled triangle.
3
5
12
52 = 5 x 5 = 25
22 = 12 x12 =144
32 = 13 x 13 =169
And so 52 + 42 = 25 + 144 =169
This is also true for the numbers 7, 24 and 25.
25
7
24
7 = 7 x 7 = 49
24 =24 x 24 = 576
25 = 25 x 25 = 625
and so 7 + 24 = 49 + 576 =25
(3,4,5), (5,12,13) and (7,24,25) are called Pythagorean triples because they satisfy the condition.
The perimeter and area of the triangle with lengths 3, 4,5 are
Perimeter = 3 + 4 + 5 = 12 units
Area = 1/2 x 3 x 4 = 6 square units
I have put these Pythagorean triples (3,4,5), (5,12,13) and (7,24,25) into a table in which I have noticed many patterns occurring.
n
a
b
c
Perimeter
Area
3
4
5
2
6
2
5
2
3
30
30
3
7
24
25
56
84
Patterns
* 'a' increases by + 2 each term.
* 'a' is equal to term number times 2 then add 1.
* The last digit of 'b' is in the pattern of 4,2,4.
* The last digit of 'c' is in the pattern of 5,3,5.
* The square root of (b + c) = a.
* 'c' is always + 1 to 'b'.
* 'b' increases by + 4 each term
* ('a' x 'n') + n = 'b'
From the first three Pythagorean triples, I have found formulae for the sides of a, b, c, and the perimeter and area.