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
- $c$ is the length of the hypotenuse
- $a$ and are the lengths of the two shorter sides
  - It does not matter which is labelled $a$ and which is labelled $b$

A right-angled triangle with the sides labelled a, b and c

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 $c = \sqrt{a^{2} + b^{2}}$
- This is just a rearrangement of the formula $a^{2} + b^{2} = c^{2}$ to make $c$ 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 $a = \sqrt{c^{2} - b^{2}}$
- This is just a rearrangement of the formula $a^{2} + b^{2} = c^{2}$ to make $a$ the subject
- Note that when finding one of the shorter sides you subtract inside the square root

Exam 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:
$A B = 12 cm$
$A C$ is a straight line, with $A D = 9 cm$ and $A C = 22 cm$

Back to Back Right Angled Triangles, IGCSE & GCSE Maths revision notes

Find $x$ , the length of side $B C .$ Give your answer to 1 decimal place.

To find $x$ , we first need to find the length of $B D$ using triangle $A B D$
Note that $B D$ is a shorter side
Apply Pythagoras' theorem, $a = \sqrt{c^{2} - b^{2}}$

$B D = \sqrt{12^{2} - 9^{2}} = \sqrt{63} = 7.93725 . . .$

It is best to leave rounding until the very end, use $\sqrt{63}$ (or $3 \sqrt{7}$ if this is what your calculator has given you) in subsequent working

Find the length of $D C$ by subtracting the length of $A D$ from the length of $A C$

$D C = 22 - 9 = 13 cm$

Now we can find $x$ using triangle $B C D$
Note that $B C$ is the hypotenuse
Apply Pythagoras' theorem, $c = \sqrt{a^{2} + b^{2}}$

$x = \sqrt{B D^{2} + D C^{2}} = \sqrt{{(\sqrt{63})}^{2} + 13^{2}} = \sqrt{63 + 169}$

$x = \sqrt{232} = 15.23154621 . . .$

Round to 1 decimal place

$x = 15.2 cm$

Pythagoras Theorem (CIE IGCSE Maths: Core)

Revision Note