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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