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Similarity

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 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

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