Pythagoras Theorem (Edexcel GCSE Maths: Higher): Revision Note

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  $a^2+b^2=c^2$

  • $c$  is the length of the hypotenuse

  • $a$  and  $b$  are the lengths of the two shorter sides 

    • It does not matter which is labelled $a$ and which is labelled $b$

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

Can I use Pythagoras’ theorem with other shapes?

• You can use Pythagoras' theorem with any shape that can be split into right-angled triangles

• You can find the length of the diagonal of a rectangle or square

  • Use its diagonal to split the shape into two identical right-angled triangles

  • The diagonal is now the hypotenuse of the right-angled triangles