Pythagorean Triads
Pythagorean Triples
A Pythagorean triple (sometimes called a Pythagorean triple) is an ordered triple $\left(a,b,c\right)$(a,b,c) of three positive integers such that $a^2+b^2=c^2$a2+b2=c2.
If $\left(a,b,c\right)$(a,b,c) is a triple then $\left(b,a,c\right)$(b,a,c) is also a triple, since $b^2+a^2$b2+a2 is the same as $a^2+b^2$a2+b2. So the order of the first two numbers in the triple doesn't matter.
If $a$a, $b$b and $c$c are relatively prime (that is, they have no common factors), then the triple is called primitive. There are $16$16 primitive Pythagorean triples with $c\le100$c≤100, including $\left(3,4,5\right)$(3,4,5), $\left(5,12,13\right)$(5,12,13), $\left(8,15,17\right)$(8,15,17), and $\left(7,24,25\right)$(7,24,25).
$\left(6,8,10\right)$(6,8,10) is also a Pythagorean triple, but it is not a primitive Pythagorean triple, since $6$6, $8$8 and $10$10 have a common factor of $2$2. If we divide each number in the triple by this common factor, we get the primitive triple $\left(3,4,5\right)$(3,4,5).
A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and will always be a right-angled triangle.
Example
Question 1
The two smallest numbers in a Pythagorean triple are $20$20 and $21$21. What number, $c$c, will complete the triple?
Think: Here we have $a=20$a=20 and $b=21$b=21 We can use the Pythagorean formula and solve for $c$c.
Do: Using the pythagorean formula,
| $c^2$c2 | $=$= | $a^2+b^2$a2+b2 |
| $=$= | $20^2+21^2$202+212 | |
| $=$= | $400+441$400+441 | |
| $=$= | $841$841 | |
| so, $c$c | $=$= | $\sqrt{841}$√841 |
| $=$= | $29$29 |
The missing value is $c=29$c=29, forming the triple $\left(20,21,29\right)$(20,21,29).
Question 2
Sean knows the two largest numbers in a Pythagorean Triple, which are $41$41 and $40$40. What number, $a$a, does Sean need to complete the triple?
Question 3
We would like to find the hypotenuse in a right-angled triangle with shorter side lengths $6$6 and $8$8, using our knowledge of common Pythagorean triples. Below are some common Pythagorean triples. The two shorter sides $6$6, $8$8 and its hypotenuse will be multiples of the sides in which of the triples? They will be multiples of the Pythagorean triple: ($\editable{}$,$\editable{}$,$\editable{}$)
What number when multiplied by $3$3 and $4$4 gives $6$6 and $8$8 respectively?
$\editable{}$ Hence, what is the length of the hypotenuse in the triangle with two shorter sides $6$6 and $8$8? $\editable{}$
A $\left(3,4,5\right)$(3,4,5)
B $\left(5,12,13\right)$(5,12,13)
C $\left(8,15,17\right)$(8,15,17)
D $\left(7,24,25\right)$(7,24,25)
Question 4
Is the Pythagorean triple $\left(66,112,130\right)$(66,112,130) a primitive Pythagorean triple?
Think: A primitive triple has no common factors between its elements. So we will need to check if $66$66, $112$112 and $130$130 have any common factors.
Do: $66$66, $112$112 and $130$130 are all even, so they have a common factor of at least $2$2. This means that $\left(66,112,130\right)$(66,112,130) is not a primitive triple.