1. A B C D
Similarity
Similar shapes are always in proportion to each other. There is no distortion between them.

2. All regular polygons are similar
Conditions for similarity
Two shapes are similar only when:
Corresponding sides are in proportion and
Corresponding angles are equal
All regular polygons are similar

4. Conditions for similarity
Two shapes are similar only when:
Corresponding sides are in proportion and
Corresponding angles are equal
All rectangles are not similar to one another since only condition 2 is true.

5. If two objects are similar then one is an enlargement of the other
The rectangles below are similar:
Find the scale factor of enlargement that maps A to B A B
8 cm
16 cm
5 cm
10 cm
Not to scale!
Scale factor = x2
Note that B to A
would be x ½

6. If two objects are similar then one is an enlargement of the other
The rectangles below are similar:
Find the scale factor of enlargement that maps A to B A B
8 cm
12 cm
5 cm
7½ cm
Not to scale!
Scale factor = x1½
Note that B to A
would be x 2/3

7. The 3 rectangles are similar. Find the unknown sides
8 cm
If we are told that two objects are similar and we can find the scale factor of enlargement then we can calculate the value of an unknown side. A B C
3 cm
Not to scale!
24 cm
x cm
y cm
13½ cm
The 3 rectangles are similar. Find the unknown sides
Comparing corresponding sides in A and B: 24/8 = 3 so x = 3 x 3 = 9 cm
Comparing corresponding sides in A and C: 13½ /3 = 4½ so y = 4 ½ x 8 = 36 cm

8. The 3 rectangles are similar. Find the unknown sides
5.6 cm
If we are told that two objects are similar and we can find the scale factor of enlargement then we can calculate the value of an unknown side. A B C
2.1 cm
Not to scale!
x cm
7.14 cm
26.88 cm
y cm
The 3 rectangles are similar. Find the unknown sides
Comparing corresponding sides in A and B: 7.14/2.1 = 3.4 so x = 3.4 x 5.6 = cm
Comparing corresponding sides in A and C: 26.88/5.6 = 4.8 so y = 4.8 x 2.1 = cm

9. Similar Triangles
Similar triangles are important in mathematics and their application can be used to solve a wide variety of problems.
The two conditions for similarity between shapes as we have seen earlier are:
Corresponding sides are in proportion and
Corresponding angles are equal
Triangles are the exception to this rule.
only the second condition is needed
Two triangles are similar if their

10. 70o
45o
65o
These two triangles are similar since they are equiangular.
50o
55o
75o
These two triangles are similar since they are equiangular.
If 2 triangles have 2 angles the same then they must be equiangular
= 180 – 125 = 55

11. Finding Unknown sides (1)
20 cm
18 cm
12 cm
6 cm b c
Since the triangles are equiangular they are similar.
So comparing corresponding sides.

12. 24 cm
10 cm
8 cm
6 cm x y
Comparing corresponding sides

13. Determining similarityB E D
Triangles ABC and DEC are similar. Why? C
Angle ABC = angle DEC (Alt angles)
Angle BAC = angle EDC (Alt angles)
Angle ACB = angle ECD (Vertically Opposite)
Since
ABC
is similar to
DEC
we know that corresponding sides are in proportion
ABDE
BCEC
ACDC
The order of the lettering is important in order to show which pairs of sides correspond.

14. A B C D E
If BC is parallel to DE, explain why triangles ABC and ADE are similar
Angle BAC = angle DAE (common to both triangles)
Angle ABC = angle ADE (corresponding angles between parallels)
Angle ACB = angle AED (corresponding angles between parallels)
A line drawn parallel to any side of a triangle produces 2 similar triangles. A D E B C B C
Triangles EBC and EAD are similar
Triangles DBC and DAE are similar

15. A B C D E
In the diagram below BE is parallel to CD and all measurements are as shown.
Calculate the length CD
Calculate the perimeter of the Trapezium EBCD
4.8 m
6 m
4 m
4.5 m
10 m A C D
7.5 cm
3 cm
8 cm
So perimeter =
= 19.8 cm

16. Alternative approach to the same problemD E
In the diagram below BE is parallel to CD and all measurements are as shown.
Calculate the length CD
Calculate the perimeter of the Trapezium EBCD
4.8 m
6 m
4 m
4.5 m
Alternative approach to the same problem
10 m A C D
7.5 cm
3 cm
8 cm
SF = 10/6 =
So CD = 1.666… x 4.8 = 8 cm
AC = 1.666… x 4.5 = 7.5
So BC = 3 cm
Perimeter = = 19.8 cm

17. The two triangles below are similar: Find the distance y.
20 cm y A E D
5 cm
45 cm

18. Alternate approach to the same problemD E
20 cm
45 cm
5 cm y
The two triangles below are similar: Find the distance y.
Alternate approach to the same problem
Scale Factor = 50/5 = 10
So y = 20/10 = 2 cm