Topic outline

  • Outcomes

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    In this course you will learn about: 

    1. The relationship between the sides of a right-angled triangle.
    2. Using the Theorem of Pythagoras to find missing lengths in a right angled triangles. 

  • Sides of a right-angled triangle

    Pythagoras was a Greek mathematician who developed one of the most famous theorems, which relates the sides of a right-angled triangle, called Pythagoras’ theorem. 

    Before you continue with Pythagoras’ theorem revise the basics you need to know about triangles by clicking on: Classifying triangles.

    A right-angled triangle has one \({90}^\circ\) angle. The longest side of the right-angled triangle is found opposite the right-angle and is called the hypotenuse (pronounced ‘high - pot - eh - news’). 

    Right-angled triangle ABC

    In the next activity you will discover the Theorem of Pythagoras for yourself.

    • Investigating the sides of a right-angled triangle

      In this activity we investigate squares on the sides of a right-angled triangle.

      Time required: 15 minutes

      What you will need:  Pen and paper

      What you will do: 

      Below is a right-angled triangle with squares on each of its sides.

      Activity image, investigating squares on the sides of a right angled triangle

      1. Write down the areas of: Square A, Square B and Square C.
      2. Add the area of square B and C. What do you get?
      3. What do you notice about the areas of B plus C compared to A?
      4. What is the length of the hypotenuse?
      5. What is the length of each of the shorter sides of the triangle?
      6. Write an equation that relates the sum of the shorter sides of the triangle to the hypotenuse.

      What did you find?

      1. By counting you will get the area of A is 25 square units.  The area of B is 16 square units.  The area of C is 9 square units. 
      2. Area of square B \(+\) Area of square C \(=16+9=25\)
      3. The areas of the sum of squares B and C is the same as the area of square A.
      4. The hypotenuse measures 5 units.
      5. The other two sides measure 4 and 3 units.
      6. \({4}^{2}+{3}^{2}={5}^{2}\).
      In the activity, you have discovered Pythagoras’ Theorem for right-angled triangles. You can also watch a video of this proof below. 

    • Theorem of Pythagoras

      In a right-angled triangle, a square formed on the hypotenuse will have the same area as the sum of the area of the two squares formed on the other sides of the triangle. 

      Right-angled triangle ABC

      If \(\Delta ABC\) is right-angled at angle B then \({AC}^{2}={AB}^{2}+{BC}^{2}\) . 

      Conversely, if  \({AC}^{2}={AB}^{2}+{BC}^{2}\) then \(\Delta ABC\) is right-angled with \(\angle B={90}^\circ\) . 

      Remember the theorem of Pythagoras can only be used in a right-angled triangle.


    • H5P icon
      Is it a right-angled triangle? Interactive Content

  • Using the Theorem of Pythagoras

    You can use the Theorem of Pythagoras to find the lengths of missing sides if you know that a triangle is right-angled. The theorem of Pythagoras has many applications in Trigonometry. 

    In the next video you will see worked examples to calculate the length of an unknown side in a right-angled triangle. This video will help you complete the next exercise.