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
Examiner 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
Use the two lengths (15 cm and 6 cm) to find the scale factor.
Multiply this by the width of the smaller rectangle to see if it applies to the width as well.
The two rectangles are similar, with a scale factor of 2.5
Show that triangles ABX and CDX are similar.
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