Pythagoras' Theorem (SQA National 5 Applications of Mathematics): Revision Note

Pythagoras' theorem

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  

  •   is the length of the hypotenuse

  •   and    are the lengths of the two shorter sides 

    • It does not matter which is labelled and which is labelled 

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

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

How do I use Pythagoras’ theorem in multi-step problems?

• You will be expected to use Pythagoras' theorem in multi-step problems such as

  • finding perimeters and areas

  • finding lengths of compound shapes involving two right-angled triangles

  • finding diameters of circles

  • finding the perpendicular height of an isosceles triangle

• The key steps for any problem are:

  • identify a right-angled triangle where you know the lengths of two of the sides

    • you might need to draw a line to split a shape into two right-angled triangles

  • use the theorem to find the third side of the triangle

  • use the length to solve the problem