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GCSEMathsAQARevision Notes4. Geometry & MeasuresRight-Angled Triangles - Pythagoras & TrigonometryPythagoras Theorem

Pythagoras Theorem (AQA GCSE Maths)

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Pythagoras Theorem

What 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 states that the longest side in a right-angled triangle is called the hypotenuse
    • The hypotenuse will always be the side opposite the right angle
  • If we label the hypotenuse c, and label the other two sides a and b then Pythagoras’ Theorem tells us that:
    • a2 + b2 = c2

      Where a, b, and c are the lengths of the three sides

Right Angled Triangle abc, IGCSE & GCSE Maths revision notes

How do I use Pythagoras’ theorem?

  • To find the length of the hypotenuse use c equals square root of a squared plus b squared end root
    • This is just a rearrangement of the formula a2 + b2 = c2, to make c the subject
    • Note that when finding the hypotenuse you add inside the square root
  • To find the length of one of the other sides use a equals square root of c squared minus b squared end root or b equals square root of c squared minus a squared end root
    • This is a rearrangement of the formula a2 + b2 = c2, to make either a or b the subject
    • Note that when finding one of the shorter sides you subtract inside the square root

Examiner Tip

  • ‘Reality check’ your answers, if the hypotenuse ends up being shorter than another side in your answer, you’ve made a mistake somewhere
  • If your calculator gives you a ‘Math ERROR’ result, you probably subtracted the wrong way around in the square root
  • In questions with multiple steps, don’t round off (to 1 decimal place, 3 significant figures, etc) until the very end

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

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 in subsequent working

Now we can find x using triangle B C D

D C space equals space 22 space minus space 9 space equals space 13 space cm

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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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.