## What is Pythagoras' theorem

In relation to Mathematics, Pythagoras’ theorem is the relation of the three sides of a right-angled triangle. The theorem can be expressed as the equation below, and relates to the lengths of the sides of the triangle.

**c² = a² + b²**

“a” & “b” are the 2 shorter sides and “c” is the Hyptoenuse (longest side)

Pythagoras’ theorem can also be rearranged to find a shorter side. For example:

a² = c² -b² **or** b² = c² – a²

Example 1:

For the triangle below, calculate the length of the Hypotenuse, x, correct to one decimal place.

Step 1: Label the sides:

a=4, b=7, c=x

Step 2: Substitute the values in Pythagoras’ theorem:

c² = a² + b²

x² = 4² + 7²

x² = 16 + 49

x = √65

x = 8.1

Example 2:

You can also use Pythagoras’ theorem to work out if the triangle below has a right angle.

Step 1:

Does c² = a² + b² ?

Step 2:

25² = 7² + 24²

625 = 49 + 576

Step 3:

625 = 625

Both sides of the equation are equal, so it is definitely a right angled triangle!

## Pythagorean Triads

On certain occasions, all 3 sides of a right-angled triangle will be whole numbers. This is called a Pythagorean Triad (also called a Pythagorean Triple). The right-angled triangle below is an example of a Pythagorean Triad.

We can use the following formula to create a Pythagorean Triad. Firstly, we need to find the middle number “m” of a Pythagorean Triad, where “s” is an odd number. The third number can then be found using Pythagoras Therum

m = (s² – 1) /2

How to create a Pythagorean Triad

Step 1: Choose any odd number for your “s” value and then square it:

7² = 49

Step 2: Find 2 consecutive numbers that add up to the squared value in “step 1”.

24 + 25 = 49

Step 3: Enter this number into the equation:

m = (7² – 1) /2

m = 24

Step 4: Write down the number you squared in step 1 and the 2 numbers from step 2. You now have your triad:

7, 24, 25

Step 5: You can use Pythagoras to check the result:

c² = a² + b²

25² = 24² + 7²

625 = 576 + 49

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