8.02 Pythagorean triples
Pythagorean triple
The numbers $3$3, $4$4 and $5$5 have a special property. If we square the first two numbers and add them they will be equal to the square of the largest number.
| $3^2+4^2$32+42 | $=$= | $5^2$52 |
| $9+16$9+16 | $=$= | $25$25 |
| $25$25 | $=$= | $25$25 |
$a^2+b^2=c^2$a2+b2=c2
That is if we substitute $a$a with $3$3, $b$b with $4$4, and $c$c with $5$5 both sides will be equal. Any group of three whole numbers that satisfy the equation are called a Pythagorean triple or a Pythagorean triad. We can check any three numbers by substituting the two smaller numbers for $a$a and $b$b and the largest number for $c$c.
Below are three Pythagorean triples:
| $\left(3,4,5\right)$(3,4,5) | $\left(5,12,13\right)$(5,12,13) | $\left(8,15,17\right)$(8,15,17) |
If you know one of the triples you can make another one by multiplying each number by a constant. For example if we multiply the triple $\left(3,4,5\right)$(3,4,5) by $2$2 we will have $\left(6,8,10\right)$(6,8,10).
$6^2+8^2=10^2$62+82=102
is also true so it will be a Pythagorean triple. The triples introduced above are now shown below with two of their multiples in each column:
| $\left(3,4,5\right)$(3,4,5) | $\left(5,12,13\right)$(5,12,13) | $\left(8,15,17\right)$(8,15,17) |
| $\left(6,8,10\right)$(6,8,10) | $\left(10,24,26\right)$(10,24,26) | $\left(16,30,34\right)$(16,30,34) |
| $\left(30,40,50\right)$(30,40,50) | $\left(50,120,130\right)$(50,120,130) | $\left(80,150,170\right)$(80,150,170) |
The three numbers of a triple are often given from smallest to largest, however sometimes it may have the first two numbers swapped, and sometimes it is in no particular order. As long as you know which number is the largest you can check if three numbers are a Pythagoran triple.
$\left(3,4,5\right)$(3,4,5), $\left(4,3,5\right)$(4,3,5), $\left(5,3,4\right)$(5,3,4)
The biggest number is $5$5 in each case, this will be $c$c.
Worked example
Example 1
Is $\left(5,7,12\right)$(5,7,12) a Pythagorean triple?
Think: A Pythagorean triple must satisfy $a^2+b^2=c^2$a2+b2=c2.
Do: Substitute $5$5 and $7$7 into $a^2+b^2$a2+b2 and see if it is equal to substituting $12$12 into $c^2$c2:
| $a^2+b^2$a2+b2 | $=$= | $5^2+7^2$52+72 |
Substsituting for left-hand side |
| $$ | $=$= | $25+49$25+49 |
|
| $$ | $=$= | $74$74 |
Evaluating |
| $c^2$c2 | $=$= | $12^2$122 |
Substsituting for right-hand side |
| $$ | $=$= | $144$144 |
Evaluating |
| $74$74 | $\ne$≠ | $144$144 |
Comparing the two sides |
| $a^2+b^2$a2+b2 | $\ne$≠ | $c^2$c2 |
|
The numbers $\left(5,7,12\right)$(5,7,12) are not a Pythagorean triple.
Reflect: We couldn't assume $a^2+b^2=c^2$a2+b2=c2 was true. We had to test each side separately and then see if they were equal.
A Pythagorean triple is any three whole numbers that satisfy
$a^2+b^2=c^2$a2+b2=c2
where $c$c is the largest number.
Right-angled triangles
In a right-angled triangle the largest angle in the triangle is $90^\circ$90°. The side across from the right angle will be the largest side. We call this side the hypotenuse.
All three sides of a right-angled triangle are related by the equation shown below:
The two smaller sides will be called $a$a and $b$b, and the hypotenuse (the longest side) will be $c$c. Earlier we looked at Pythagorean triples which satisfy the same equation. Any triangle with sides that are a Pythagorean triple will be a right-angled triangle.
Worked example
Example 2
Is the following triangle a right-angled triangle?
Think: If its three side lengths satisfy $a^2+b^2=c^2$a2+b2=c2 then the triangle will be a right-angled triangle.
Do:
| $a^2+b^2$a2+b2 | $=$= | $15^2+20^2$152+202 |
Calculating left-hand side |
| $$ | $=$= | $225+400$225+400 |
|
| $$ | $=$= | $625$625 |
Evaluating |
| $c^2$c2 | $=$= | $25^2$252 |
Calculating right-hand side |
| $$ | $=$= | $625$625 |
Evaluating |
| $a^2+b^2$a2+b2 | $=$= | $c^2$c2 |
|
Reflect: We can also skip some of the working out if we realise that $\left(15,20,25\right)$(15,20,25) is a multiple of $\left(3,4,5\right)$(3,4,5), because each number has been multiplied by $5$5. This means it will also be a Pythagorean triple and the triangle will be a right-angled triangle.
Pythagoras' theorem relates the three sides of a right-angled triangle, $a$a and $b$b are the two smaller sides, and the longest side, called the hypotenuse, is $c$c.
We can also test to see if a triangle is right-angled by checking to see if its three sides satisfy $a^2+b^2=c^2$a2+b2=c2.
Practice questions
Question 1
Using your knowledge of common Pythagorean triples or otherwise, is $\left(12,5,13\right)$(12,5,13) a Pythagorean triple?
Yes
ANo
B
Question 2
Which side of the triangle in the diagram is the hypotenuse?
$X$X
A$Y$Y
B$Z$Z
C
Question 3
Use Pythagoras' theorem to determine whether this is a right-angled triangle.
Let $a$a and $b$b represent the two shorter side lengths. First find the value of $a^2+b^2$a2+b2.
Let $c$c represent the length of the longest side. Find the value of $c^2$c2.
Is the triangle a right-angled triangle?
Yes
ANo
B