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Pythagoras Theorem (Cambridge (CIE) IGCSE International Maths)

Revision Note

Dan Finlay

Written by: Dan Finlay

Reviewed by: Lucy Kirkham

Cambridge (CIE) International Maths: ExtendedEdexcel Maths A (Modular): Higher Unit 1Cambridge (CIE) International Maths: CoreCambridge (CIE) Maths: CoreCambridge (CIE) Maths: ExtendedEdexcel Maths A: Higher

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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 Tips and Tricks

  • 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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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    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.

    Lucy Kirkham

    Author: Lucy Kirkham

    Expertise: Head of STEM

    Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels.Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all.