Steps for solving Pythagorean triplet of an odd number?
Steps for solving Pythagorean triplet of an odd number?
PYTHAGOREAN TRIPLE USING ODD NUMBERS
Pythagoras theorem
In a right-angled triangle, the square of the hypotenuse is equal to the square of the other two sides i.e
a² + b² = c²
Where a or b = adjacent or opposite
And c = hypotenuse
Pythagorean Triple
A Pythagorean triple is a set of three (3) numbers; a, b, and c that are integers such that a² + b² = c².
Odd numbers
Odd numbers are numbers that are not divisible by 2. Examples include 3, 5, 7, 9, 11, etc
Having known what Pythagorean triple and odd numbers are, let’s proceed to find out how to form Pythagorean triple using odd numbers
Step 1: Assuming a is an odd number.
Step 2: Square it. That will be equal to a²
Step 3: Subtract one (1) from the number.
a² – 1
Step 4: Divide the result by 2. That is
(a² – 1)/2
Step 5: Since The resulting expression {(a² – 1)/2} are the same, add one (1) to one of them
{(a² – 1)/2 + 1}
{(a² – 1) + 2} / 2
(a² – 1 + 2) / 2
(a² + 1) / 2
So, the three expressions are
a, (a² – 1)/2, and (a² + 1)/2
Their squares become
a², {(a² – 1)/2}², and {(a² + 1)/2}²
Thus, a² + {(a² – 1)/2}² = {(a² + 1)/2}²
Examples of odd numbers ranging from one (1) to fifty (50) are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49. We will use the first five numbers excluding one (1) to illustrate the formula
Using 1 as a number will result in having one of the sides equal to zero, which will result in a triple (1, 0, 1), but definitely not a triangle, as no sides in a triangle is equal to Zero. – Law of the null effect – Kenneth
I. Let a = 3
3² + {(3² – 1)/2}² = {(3² + 1)/2}²
3² + {(9 – 1)/2}² = {(9 + 1)/2}²
3² + (8/2)² = (10/2)²
3² + 4² = 5²
9 + 16 = 25
25 = 25
II. Let a = 5
5² + {(5² – 1)/2}² = {(5² + 1)/2}²
5² + {(25 – 1)/2}² = {(25 + 1)/2}²
5² + (24/2)² = (26/2)²
5² + 12² = 13²
25 + 144 = 169
169 = 169
III. Let a = 7
7² + {(7² – 1)/2}² = {(7² + 1)/2}²
7² + {(49 – 1)/2}² = {(49 + 1)/2}²
7² + (48/2)² = (50/2)²
7² + 24² = 25²
49 + 576 = 625
625 = 625
PYTHAGOREAN TRIPLE USING ODD NUMBERS
Pythagoras theorem
In a right-angled triangle, the square of the hypotenuse is equal to the square of the other two sides i.e
a² + b² = c²
Where a or b = adjacent or opposite
And c = hypotenuse
Pythagorean Triple
A Pythagorean triple is a set of three (3) numbers; a, b, and c that are integers such that a² + b² = c².
Odd numbers
Odd numbers are numbers that are not divisible by 2. Examples include 3, 5, 7, 9, 11, etc
Having known what Pythagorean triple and odd numbers are, let’s proceed to find out how to form Pythagorean triple using odd numbers
Step 1: Assuming a is an odd number.
Step 2: Square it. That will be equal to a²
Step 3: Subtract one (1) from the number.
a² – 1
Step 4: Divide the result by 2. That is
(a² – 1)/2
Step 5: Since The resulting expression {(a² – 1)/2} are the same, add one (1) to one of them
{(a² – 1)/2 + 1}
{(a² – 1) + 2} / 2
(a² – 1 + 2) / 2
(a² + 1) / 2
So, the three expressions are
a, (a² – 1)/2, and (a² + 1)/2
Their squares become
a², {(a² – 1)/2}², and {(a² + 1)/2}²
Thus, a² + {(a² – 1)/2}² = {(a² + 1)/2}²
Examples of odd numbers ranging from one (1) to fifty (50) are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49. We will use the first five numbers excluding one (1) to illustrate the formula
Using 1 as a number will result in having one of the sides equal to zero, which will result in a triple (1, 0, 1), but definitely not a triangle, as no sides in a triangle is equal to Zero. – Law of the null effect – Kenneth
I. Let a = 3
3² + {(3² – 1)/2}² = {(3² + 1)/2}²
3² + {(9 – 1)/2}² = {(9 + 1)/2}²
3² + (8/2)² = (10/2)²
3² + 4² = 5²
9 + 16 = 25
25 = 25
II. Let a = 5
5² + {(5² – 1)/2}² = {(5² + 1)/2}²
5² + {(25 – 1)/2}² = {(25 + 1)/2}²
5² + (24/2)² = (26/2)²
5² + 12² = 13²
25 + 144 = 169
169 = 169
III. Let a = 7
7² + {(7² – 1)/2}² = {(7² + 1)/2}²
7² + {(49 – 1)/2}² = {(49 + 1)/2}²
7² + (48/2)² = (50/2)²
7² + 24² = 25²
49 + 576 = 625
625 = 625
