Pythagoras Theorem (Edexcel GCSE Maths: Foundation)

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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 squared plus b squared equals c squared
    • 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

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 equals square root of a squared plus b squared end root
    • This is just a rearrangement of the formula a squared plus b squared equals c squared 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 equals square root of c squared minus b squared end root
    • This is just a rearrangement of the formula a squared plus b squared equals c squared to make a 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:
A B space equals space 12 space cm
A C is a straight line, with A D space equals space 9 space cm and A C space equals space 22 space 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 equals square root of c squared minus b squared end root

B D space equals space square root of 12 squared minus 9 squared end root space equals space square root of 63 space equals space 7.93725...

It is best to leave rounding until the very end, use square root of 63 (or 3 square root of 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 space equals space 22 space minus space 9 space equals space 13 space cm

Now we can find x using triangle B C D
Note that B C is the hypotenuse
Apply Pythagoras' theorem, c equals square root of a squared plus b squared end root

x equals square root of B D squared plus D C squared end root equals square root of open parentheses square root of 63 close parentheses squared plus 13 squared end root equals square root of 63 plus 169 end root

x equals square root of 232 space equals space 15.23154621...

Round to 1 decimal place

bold italic x bold equals bold 15 bold. bold 2 bold space bold cm

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.