IGCSEMaths: ExtendedCIERevision Notes3. Geometry3.8 Congruence & Similarity3.8.2 Similarity

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Similarity (CIE IGCSE Maths: Extended)

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Similarity

What are similar shapes?

  • Two shapes are similar if they have the same shape and their corresponding sides are in proportion
    • One shape is an enlargement of the other 
  • If two triangles of different sizes have 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 the scale 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

Exam Tip

  • Proving two shapes are similar can require a lot of writing, you do not need to write in full sentences, but you must make sure you quote all of the keywords to get the marks

Worked example

a)
Prove that the two rectangles shown in the diagram below are similar.
 

4-5-1-similarity-we-question

Use the two lengths (15 cm and 6 cm) to find the scale factor.

15 over 6 space equals space 5 over 2 space equals space 2.5

Multiply this by the width of the smaller rectangle to see if it applies to the width as well.

2 space cross times space 2.5 space equals space 5

The two rectangles are similar, with a scale factor of 2.5

b)
In the diagram below, AB and CD are parallel lines.
Show that triangles ABX and CDX are similar.

DtyUw7xt_4-5-1-similarity-we-question

State the equal angles by name, along with clear reasons.
Don't forget to state that similar triangles need to have equal corresponding angles.

Angle AXB = angle CXD (Vertically opposite angles are equal)
Angle ABC = angle BCD (Alternate angles on parallel lines are equal)
Angle BAD = angle ADC (Alternate angles on parallel lines are equal)

All three corresponding angles are equal, so the two triangles are similar

Similar Lengths

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

Exam Tip

  • If similar shapes overlap on the diagram (or are not clear) draw them separately
    • For example, in this diagram the triangles ABC and APQ are similar:
    • Similar Triangles together, IGCSE & GCSE Maths revision notesSo we would redraw them separately before we start:Similar-Triangles-separately, IGCSE & GCSE Maths revision notes

Worked example

ABCD and PQRS are similar shapes.
Similarity – Lengths Example shapes, IGCSE & GCSE Maths revision notesFind the length of PS.

As the two shapes are mathematically similar, there will exist a value of k such that table row cell A D space end cell equals cell space k P S end cell end table and table row cell A B space end cell equals cell space k P Q end cell end table.
k is known as the scale factor.

Form an equation using the two known corresponding sides of the triangle.
 

table attributes columnalign right center left columnspacing 0px end attributes row cell A B space end cell equals cell space k P Q end cell row cell space 6 space end cell equals cell space 3 k end cell end table

Solve to find k.

table row cell k space end cell equals cell space 6 over 3 space equals space 2 end cell end table

Substitute into table row cell A D space end cell equals cell space k P S end cell end table.

table row cell A D space end cell equals cell space 2 P S end cell row cell 15 space end cell equals cell space 2 P S end cell end table

Solve to find table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell P S end cell end table.

table row cell P S space end cell equals cell space 15 over 2 end cell end table

bold italic P bold italic S bold space bold equals bold space bold 7 bold. bold 5 bold space bold cm

Similar Areas & Volumes

What are similar shapes?

  • Two shapes are mathematically similar if one is an enlargement of the other
  • If two similar shapes are linked by the scale 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 known lengths, areas or volumes
    • 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

Exam Tip

  • Take extra care not to mix up which shape is which when you have started carrying out the calculations
  • It can help to label the shapes and always write an equation 
    • For example if shape A is similar to shape B:
      • length A = k(length B)
      • area A = k2(area B)
      • volume A = k3(volume B)

Worked example

Solid A and solid B are mathematically similar. 

The volume of solid A is 32 cm3.
The volume of solid B is 108 cm3.
The height of solid A is 10 cm.

Find the height of solid B.

Calculate k cubed, the scale factor of enlargement for the volumes, using volume space B space equals space k cubed open parentheses volume space A close parentheses,

Or k cubed equals fraction numerator larger space volume over denominator smaller space volume end fraction.

table row cell 108 space end cell equals cell space 32 k cubed end cell row cell k cubed space end cell equals cell space 108 over 32 space equals space 27 over 8 end cell end table
 

For similar shapes, if the volume scale factor is k cubed, then the length scale factor is k.
FInd k.

k space equals space cube root of 27 over 8 end root space equals space 3 over 2
 

Substitute into formula for the heights of the similar shapes. Height space B space equals space k open parentheses heigh t space A close parentheses,

table row cell h space end cell equals cell space 10 k end cell row cell h space end cell equals cell space 10 open parentheses 3 over 2 close parentheses space equals space 30 over 2 space equals space 15 end cell end table
 

Height of B = 15 cm

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    Amber

    Author: Amber

    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.