Pythagoras Theorem
Who is Pythagoras?
- Pythagoras was a Greek mathematician who lived over 2500 years ago
- He is most famous for Pythagoras’ Theorem, which includes the important formula for right-angled triangles
What is Pythagoras' theorem?
- Pythagoras' theorem is a formula that links the lengths of the three sides of a right-angled triangle
- The longest side of a right-angled triangle is called the hypotenuse
- The hypotenuse will always be the side opposite the right angle
- Pythagoras' theorem states that
- is the length of the hypotenuse
- and are the lengths of the two shorter sides
- It does not matter which is labelled and which is labelled
How do I use Pythagoras’ theorem to find the length of the hypotenuse?
- To find the length of the hypotenuse
- Square the lengths of the two shorter sides
- Add these two numbers together
- Take the positive square root
- This can be written as
- This is just a rearrangement of the formula to make the subject
- Note that when finding the hypotenuse you add inside the square root
How do I use Pythagoras’ theorem to find the length of a shorter side?
- To find the length of a shorter side
- Square the lengths of the hypotenuse and the other shorter side
- Subtract these numbers to find the difference
- Take the positive square root
- This can be written as
- This is just a rearrangement of the formula to make the subject
- Note that when finding one of the shorter sides you subtract inside the square root
Examiner Tip
- If the hypotenuse ends up being shorter than another side in your answer then you have made a mistake somewhere
- Make sure that you subtract the smaller value from the bigger value when finding the length of a shorter side
- Otherwise you will get a "Math Error" when trying to find the square root of a negative number
- In questions with multiple steps:
- Leave your answer as an exact answer
- Do not round until the very end of the question
Worked example
In the following diagram:
is a straight line, with and
Find , the length of side Give your answer to 1 decimal place.
To find , we first need to find the length of using triangle
Note that is a shorter side
Apply Pythagoras' theorem,
It is best to leave rounding until the very end, use (or if this is what your calculator has given you) in subsequent working
Find the length of by subtracting the length of from the length of
Now we can find using triangle
Note that is the hypotenuse
Apply Pythagoras' theorem,
Round to 1 decimal place