The numbers 3,\,4 and 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.\begin{aligned}3^{2}+4^{2} &= 5^{2}\\9+16&=25 \\25&=25\end{aligned}
We can also think about these numbers as satisfying the equation: a^{2}+b^{2}=c^{2}
That is if we substitute a with 3,\,b with 4, and c with 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 and b and the largest number for c.
Below are three Pythagorean triples: (3,\,4,\,5),\,\,(5,\,12,\,13),\,\,(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 (3,\,4,\,5) by 2 we will have (6,\,8,\,10). 6^{2}+8^{2}=10^{2} 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:
(3,\,4,\,5) | (5,\,12,\,13) | (8,\,15,\,17) | |
---|---|---|---|
\times 2 | (6,\,8,\,10) | (10,\,24,\,26) | (16,\,30,\,34) |
\times 10 | (30,\,40,\,50) | (50,\,120,\,130) | (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.(3,\,4,\,5),\,\,(4,\,3,\,5),\,\,(5,\,3,\,4)
The biggest number is 5 in each case, this will be c.
Is (12,\,5,\,13) a Pythagorean triple?
A Pythagorean triple is any three whole numbers that satisfy
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.
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}.
Use Pythagoras' theorem to determine whether this is a right-angled triangle.
Let a and b represent the two shorter side lengths. First find the value of a^{2}+b^{2}.
Let c represent the length of the longest side. Find the value of c^{2}.
Is the triangle a right-angled triangle?