 Lesson ## 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 $c\le100$c100, 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.

1. 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?

 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)

They will be multiples of the Pythagorean triple:

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2. What number when multiplied by $3$3 and $4$4 gives $6$6 and $8$8 respectively?

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3. Hence, what is the length of the hypotenuse in the triangle with two shorter sides $6$6 and $8$8?

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##### 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.