3.8.2 Similarity | CIE IGCSE Maths: Extended Revision Notes 2023 (2024)

What are similar shapes?

• Two shapes are similar if they have the sameshape and their corresponding sidesare inproportion
  • One shape is anenlargement of the other
• If two triangles of different sizeshave the same angles they are similar
  • Other shapes can have the same angles and not be similar

How do we prove that two shapes are similar?

• To show that two non triangular shapes are similar you need to show that their corresponding sides are in proportion
  • Divide the length of one side by the length of the same side on the other shape to find thescale factor
• If the scale factor is the same for all corresponding sides, then the shapes are similar
• If one shape can be shown to be an enlargement of the other, then the two shapes are similar

How do we prove that two triangles are similar?

• To show that two triangles are similar you simply need to show that their angles are the same
• This can be done through angle properties, look for isosceles triangles, vertically opposite angles and angles on parallel lines
• The triangles may not look similar and may be facing in different directions to each other, so concentrate on finding the angles
  • it may help to sketch both triangles next to each other and facing the same direction
• If a question asks you to prove two triangles are similar, you will need to state that corresponding angles in similar triangles are the same and you will need to give a reason for each corresponding equal angle
  • The triangles can often be opposite each other in an hourglass formation, look out for the vertically opposite, equal angles

How do I work with similar lengths?

• Equivalent lengths in two similar shapes will be in the same ratio and are linked by a scale factor
  • Normally the first step is to find this scale factor
  • STEP 1
    Identify equivalent known lengths
  • STEP 2
    Establish direction
    • If the scale factor is greater than 1 the shape is getting bigger
    • If the scale factor is less than 1 the shape is getting smaller
  • STEP 3
    Find the scale factor
    • Second Length ÷ First Length

• • STEP 4
    Use scale factor to find the length you need

What are similar shapes?

• Two shapes are mathematically similar if one is an enlargement of the other
• If two similar shapes are linked by thescale factor, k
  
  • Equivalent areas are linked by an area factor, k²
  • Equivalent volumes are linked by a volume factor, k³

How do I work with similar shapes involving area or volume?

• STEP 1
  Identify the equivalent known quantities
  • These could be for lengths, areas or volumes
• STEP 2
  Establish direction
  • Are they getting bigger or smaller?
• STEP 3
  Find the Scale Factor from two knownlengths, areasorvolumes
  • Second Quantity ÷ First Quantity
  • Check the scale factor is > 1 if getting bigger and < 1 if getting smaller
  • If the scale factor, s.f., is from two lengths, write it as k= s.f.
  • If the scale factor, s.f., is from two areas, write it as k²= s.f.
  • If the scale factor, s.f., is from two volumes, write it as k³= s.f.
• STEP 4
  Use the value of the scale factor you have found to convert other corresponding lengths, areas or volumes using
  • Area Scale Factor = (Length Scale Factor)²
    • Or Length Scale Factor = √(Area Scale Factor)
  • Volume Scale Factor = (Length Scale Factor)³
    • Or Length Scale Factor = ∛(Volume Length Factor)
• Use the scale factor to find a new quantity